Description
Smooth testing of goodness of fit. These tests are data driven (alternative hypothesis is dynamically selected based on data). In this package you will find various tests for exponent, Gaussian, Gumbel and uniform distribution.
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Maintainer: ‘Przemyslaw Biecek <przemyslaw.biecek@gmail.com>’
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Authors@R: c(person(given = "Przemyslaw",
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Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:31: Lost braces; missing escapes or markup?
31 | where \emph{$l(Z_i)$}, i=1,...,n, is \emph{k}-dimensional (row) score vector, the symbol \emph{'} denotes transposition while \emph{$I=Cov_{theta_0}[l(Z_1)]'[l(Z_1)]$}. Following Neyman's idea of modelling underlying distributions one gets \emph{$l(Z_i)=(phi_1(F(Z_i)),...,phi_k(F(Z_i)))$} and \emph{I} being the identity matrix, where \emph{$phi_j$}'s, j >= 1, are zero mean orthonormal functions on [0,1], while \emph{F} is the completely specified null distribution function.
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:36: Lost braces; missing escapes or markup?
36 | where \emph{$tilde gamma$} is an appropriate estimator of \emph{$gamma$} while \emph{$I^*(gamma)=Cov_{theta_0}[l^*(Z_1;gamma)]'[l^*(Z_1;gamma)]$}. More details can be found in Janic and Ledwina (2008), Kallenberg and Ledwina (1997 a,b) as well as Inglot and Ledwina (2006 a,b).
| ^
checkRd: (-1) ddst-package.Rd:40: Lost braces
40 | \emph{$T = min{1 <= k <= d: W_k-pi(k,n,c) >= W_j-pi(j,n,c), j=1,...,d}$}
| ^
checkRd: (-1) ddst-package.Rd:45: Lost braces
45 | $T^* = min{1 <= k <= d: W_k^*(tilde gamma)-pi^*(k,n,c) >= W_j^*(tilde gamma)-pi^*(j,n,c), j=1,...,d}$}.
| ^
checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
| ^
checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
| ^
checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
| ^
checkRd: (-1) ddst-package.Rd:54: Lost braces
54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
| ^
checkRd: (-1) ddst-package.Rd:54: Lost braces
54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
| ^
checkRd: (-1) ddst-package.Rd:58: Lost braces; missing escapes or markup?
58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
| ^
checkRd: (-1) ddst-package.Rd:58: Lost braces; missing escapes or markup?
58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
| ^
checkRd: (-1) ddst-package.Rd:62: Lost braces; missing escapes or markup?
62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
| ^
checkRd: (-1) ddst-package.Rd:62: Lost braces; missing escapes or markup?
62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
| ^
checkRd: (-1) ddst-package.Rd:65: Lost braces; missing escapes or markup?
65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
| ^
checkRd: (-1) ddst-package.Rd:65: Lost braces; missing escapes or markup?
65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
| ^
checkRd: (-1) ddst-package.Rd:67: Lost braces; missing escapes or markup?
67 | The choice of \emph{c} in \emph{T} and \emph{$T^*$} is decisive to finite sample behaviour of the selection rules and pertaining statistics \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$}. In particular, under large \emph{c}'s the rules behave similarly as Schwarz's (1978) BIC while for \emph{c=0} they mimic Akaike's (1973) AIC. For moderate sample sizes, values \emph{c in (2,2.5)} guarantee, under `smooth' departures, only slightly smaller power as in case BIC were used and simultaneously give much higher power than BIC under multimodal alternatives. In genral, large \emph{c's} are recommended if changes in location, scale, skewness and kurtosis are in principle aimed to be detected. For evidence and discussion see Inglot and Ledwina (2006 c).
| ^
checkRd: (-1) ddst-package.Rd:69: Lost braces; missing escapes or markup?
69 | It \emph{c>0} then the limiting null distribution of \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} is central chi-squared with one degree of freedom. In our implementation, for given \emph{n}, both critical values and \emph{p}-values are computed by MC method.
| ^
checkRd: (-1) ddst-package.Rd:71: Lost braces; missing escapes or markup?
71 | Empirical distributions of \emph{T} and \emph{$T^*$} as well as \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} are not essentially influenced by the choice of reasonably large \emph{d}'s, provided that sample size is at least moderate.
| ^
checkRd: (-1) ddst.exp.test.Rd:27: Lost braces; missing escapes or markup?
27 | Modelling alternatives similarly as in Kallenberg and Ledwina (1997 a,b), e.g., and estimating \emph{$gamma$} by \emph{$tilde gamma= 1/n sum_{i=1}^n Z_i$} yields the efficient score
| ^
checkRd: (-1) ddst.exp.test.Rd:30: Lost braces; missing escapes or markup?
30 | The matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and computed in a numerical way in case of cosine basis. In the implementation the default value of \emph{c} in \emph{$T^*$} is set to be 100.
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:30: Lost braces; missing escapes or markup?
30 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=1/n sum_{i=1}^n Z_i$} and
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.uniform.test.Rd:25: Lost braces; missing escapes or markup?
25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
| ^
checkRd: (-1) ddst.uniform.test.Rd:25: Lost braces; missing escapes or markup?
25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
| ^
NOTE
r-devel-linux-x86_64-debian-gcc
CRAN incoming feasibility
Maintainer: ‘Przemyslaw Biecek <przemyslaw.biecek@gmail.com>’
No Authors@R field in DESCRIPTION.
Please add one, modifying
Authors@R: c(person(given = "Przemyslaw",
family = "Biecek",
role = c("aut", "cre"),
email = "przemyslaw.biecek@gmail.com",
comment = "R code"),
person(given = "Teresa",
family = "Ledwina",
role = "aut",
comment = "support,\n descriptions"))
as necessary.
NOTE
r-devel-linux-x86_64-debian-gcc
Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:31: Lost braces; missing escapes or markup?
31 | where \emph{$l(Z_i)$}, i=1,...,n, is \emph{k}-dimensional (row) score vector, the symbol \emph{'} denotes transposition while \emph{$I=Cov_{theta_0}[l(Z_1)]'[l(Z_1)]$}. Following Neyman's idea of modelling underlying distributions one gets \emph{$l(Z_i)=(phi_1(F(Z_i)),...,phi_k(F(Z_i)))$} and \emph{I} being the identity matrix, where \emph{$phi_j$}'s, j >= 1, are zero mean orthonormal functions on [0,1], while \emph{F} is the completely specified null distribution function.
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:36: Lost braces; missing escapes or markup?
36 | where \emph{$tilde gamma$} is an appropriate estimator of \emph{$gamma$} while \emph{$I^*(gamma)=Cov_{theta_0}[l^*(Z_1;gamma)]'[l^*(Z_1;gamma)]$}. More details can be found in Janic and Ledwina (2008), Kallenberg and Ledwina (1997 a,b) as well as Inglot and Ledwina (2006 a,b).
| ^
checkRd: (-1) ddst-package.Rd:40: Lost braces
40 | \emph{$T = min{1 <= k <= d: W_k-pi(k,n,c) >= W_j-pi(j,n,c), j=1,...,d}$}
| ^
checkRd: (-1) ddst-package.Rd:45: Lost braces
45 | $T^* = min{1 <= k <= d: W_k^*(tilde gamma)-pi^*(k,n,c) >= W_j^*(tilde gamma)-pi^*(j,n,c), j=1,...,d}$}.
| ^
checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
| ^
checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
| ^
checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
| ^
checkRd: (-1) ddst-package.Rd:54: Lost braces
54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
| ^
checkRd: (-1) ddst-package.Rd:54: Lost braces
54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
| ^
checkRd: (-1) ddst-package.Rd:58: Lost braces; missing escapes or markup?
58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
| ^
checkRd: (-1) ddst-package.Rd:58: Lost braces; missing escapes or markup?
58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
| ^
checkRd: (-1) ddst-package.Rd:62: Lost braces; missing escapes or markup?
62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
| ^
checkRd: (-1) ddst-package.Rd:62: Lost braces; missing escapes or markup?
62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
| ^
checkRd: (-1) ddst-package.Rd:65: Lost braces; missing escapes or markup?
65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
| ^
checkRd: (-1) ddst-package.Rd:65: Lost braces; missing escapes or markup?
65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
| ^
checkRd: (-1) ddst-package.Rd:67: Lost braces; missing escapes or markup?
67 | The choice of \emph{c} in \emph{T} and \emph{$T^*$} is decisive to finite sample behaviour of the selection rules and pertaining statistics \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$}. In particular, under large \emph{c}'s the rules behave similarly as Schwarz's (1978) BIC while for \emph{c=0} they mimic Akaike's (1973) AIC. For moderate sample sizes, values \emph{c in (2,2.5)} guarantee, under `smooth' departures, only slightly smaller power as in case BIC were used and simultaneously give much higher power than BIC under multimodal alternatives. In genral, large \emph{c's} are recommended if changes in location, scale, skewness and kurtosis are in principle aimed to be detected. For evidence and discussion see Inglot and Ledwina (2006 c).
| ^
checkRd: (-1) ddst-package.Rd:69: Lost braces; missing escapes or markup?
69 | It \emph{c>0} then the limiting null distribution of \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} is central chi-squared with one degree of freedom. In our implementation, for given \emph{n}, both critical values and \emph{p}-values are computed by MC method.
| ^
checkRd: (-1) ddst-package.Rd:71: Lost braces; missing escapes or markup?
71 | Empirical distributions of \emph{T} and \emph{$T^*$} as well as \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} are not essentially influenced by the choice of reasonably large \emph{d}'s, provided that sample size is at least moderate.
| ^
checkRd: (-1) ddst.exp.test.Rd:27: Lost braces; missing escapes or markup?
27 | Modelling alternatives similarly as in Kallenberg and Ledwina (1997 a,b), e.g., and estimating \emph{$gamma$} by \emph{$tilde gamma= 1/n sum_{i=1}^n Z_i$} yields the efficient score
| ^
checkRd: (-1) ddst.exp.test.Rd:30: Lost braces; missing escapes or markup?
30 | The matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and computed in a numerical way in case of cosine basis. In the implementation the default value of \emph{c} in \emph{$T^*$} is set to be 100.
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:30: Lost braces; missing escapes or markup?
30 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=1/n sum_{i=1}^n Z_i$} and
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.uniform.test.Rd:25: Lost braces; missing escapes or markup?
25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
| ^
checkRd: (-1) ddst.uniform.test.Rd:25: Lost braces; missing escapes or markup?
25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
| ^
NOTE
r-devel-linux-x86_64-fedora-clang
Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:31: Lost braces; missing escapes or markup?
31 | where \emph{$l(Z_i)$}, i=1,...,n, is \emph{k}-dimensional (row) score vector, the symbol \emph{'} denotes transposition while \emph{$I=Cov_{theta_0}[l(Z_1)]'[l(Z_1)]$}. Following Neyman's idea of modelling underlying distributions one gets \emph{$l(Z_i)=(phi_1(F(Z_i)),...,phi_k(F(Z_i)))$} and \emph{I} being the identity matrix, where \emph{$phi_j$}'s, j >= 1, are zero mean orthonormal functions on [0,1], while \emph{F} is the completely specified null distribution function.
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:36: Lost braces; missing escapes or markup?
36 | where \emph{$tilde gamma$} is an appropriate estimator of \emph{$gamma$} while \emph{$I^*(gamma)=Cov_{theta_0}[l^*(Z_1;gamma)]'[l^*(Z_1;gamma)]$}. More details can be found in Janic and Ledwina (2008), Kallenberg and Ledwina (1997 a,b) as well as Inglot and Ledwina (2006 a,b).
| ^
checkRd: (-1) ddst-package.Rd:40: Lost braces
40 | \emph{$T = min{1 <= k <= d: W_k-pi(k,n,c) >= W_j-pi(j,n,c), j=1,...,d}$}
| ^
checkRd: (-1) ddst-package.Rd:45: Lost braces
45 | $T^* = min{1 <= k <= d: W_k^*(tilde gamma)-pi^*(k,n,c) >= W_j^*(tilde gamma)-pi^*(j,n,c), j=1,...,d}$}.
| ^
checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
| ^
checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
| ^
checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
| ^
checkRd: (-1) ddst-package.Rd:54: Lost braces
54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
| ^
checkRd: (-1) ddst-package.Rd:54: Lost braces
54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
| ^
checkRd: (-1) ddst-package.Rd:58: Lost braces; missing escapes or markup?
58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
| ^
checkRd: (-1) ddst-package.Rd:58: Lost braces; missing escapes or markup?
58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
| ^
checkRd: (-1) ddst-package.Rd:62: Lost braces; missing escapes or markup?
62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
| ^
checkRd: (-1) ddst-package.Rd:62: Lost braces; missing escapes or markup?
62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
| ^
checkRd: (-1) ddst-package.Rd:65: Lost braces; missing escapes or markup?
65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
| ^
checkRd: (-1) ddst-package.Rd:65: Lost braces; missing escapes or markup?
65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
| ^
checkRd: (-1) ddst-package.Rd:67: Lost braces; missing escapes or markup?
67 | The choice of \emph{c} in \emph{T} and \emph{$T^*$} is decisive to finite sample behaviour of the selection rules and pertaining statistics \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$}. In particular, under large \emph{c}'s the rules behave similarly as Schwarz's (1978) BIC while for \emph{c=0} they mimic Akaike's (1973) AIC. For moderate sample sizes, values \emph{c in (2,2.5)} guarantee, under `smooth' departures, only slightly smaller power as in case BIC were used and simultaneously give much higher power than BIC under multimodal alternatives. In genral, large \emph{c's} are recommended if changes in location, scale, skewness and kurtosis are in principle aimed to be detected. For evidence and discussion see Inglot and Ledwina (2006 c).
| ^
checkRd: (-1) ddst-package.Rd:69: Lost braces; missing escapes or markup?
69 | It \emph{c>0} then the limiting null distribution of \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} is central chi-squared with one degree of freedom. In our implementation, for given \emph{n}, both critical values and \emph{p}-values are computed by MC method.
| ^
checkRd: (-1) ddst-package.Rd:71: Lost braces; missing escapes or markup?
71 | Empirical distributions of \emph{T} and \emph{$T^*$} as well as \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} are not essentially influenced by the choice of reasonably large \emph{d}'s, provided that sample size is at least moderate.
| ^
checkRd: (-1) ddst.exp.test.Rd:27: Lost braces; missing escapes or markup?
27 | Modelling alternatives similarly as in Kallenberg and Ledwina (1997 a,b), e.g., and estimating \emph{$gamma$} by \emph{$tilde gamma= 1/n sum_{i=1}^n Z_i$} yields the efficient score
| ^
checkRd: (-1) ddst.exp.test.Rd:30: Lost braces; missing escapes or markup?
30 | The matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and computed in a numerical way in case of cosine basis. In the implementation the default value of \emph{c} in \emph{$T^*$} is set to be 100.
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:30: Lost braces; missing escapes or markup?
30 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=1/n sum_{i=1}^n Z_i$} and
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
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checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.uniform.test.Rd:25: Lost braces; missing escapes or markup?
25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
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checkRd: (-1) ddst.uniform.test.Rd:25: Lost braces; missing escapes or markup?
25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
| ^
NOTE
r-devel-linux-x86_64-fedora-gcc
Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
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checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
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checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
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checkRd: (-1) ddst-package.Rd:31: Lost braces; missing escapes or markup?
31 | where \emph{$l(Z_i)$}, i=1,...,n, is \emph{k}-dimensional (row) score vector, the symbol \emph{'} denotes transposition while \emph{$I=Cov_{theta_0}[l(Z_1)]'[l(Z_1)]$}. Following Neyman's idea of modelling underlying distributions one gets \emph{$l(Z_i)=(phi_1(F(Z_i)),...,phi_k(F(Z_i)))$} and \emph{I} being the identity matrix, where \emph{$phi_j$}'s, j >= 1, are zero mean orthonormal functions on [0,1], while \emph{F} is the completely specified null distribution function.
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checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
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checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
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checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
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checkRd: (-1) ddst-package.Rd:36: Lost braces; missing escapes or markup?
36 | where \emph{$tilde gamma$} is an appropriate estimator of \emph{$gamma$} while \emph{$I^*(gamma)=Cov_{theta_0}[l^*(Z_1;gamma)]'[l^*(Z_1;gamma)]$}. More details can be found in Janic and Ledwina (2008), Kallenberg and Ledwina (1997 a,b) as well as Inglot and Ledwina (2006 a,b).
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checkRd: (-1) ddst-package.Rd:40: Lost braces
40 | \emph{$T = min{1 <= k <= d: W_k-pi(k,n,c) >= W_j-pi(j,n,c), j=1,...,d}$}
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45 | $T^* = min{1 <= k <= d: W_k^*(tilde gamma)-pi^*(k,n,c) >= W_j^*(tilde gamma)-pi^*(j,n,c), j=1,...,d}$}.
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checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
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checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
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checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
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checkRd: (-1) ddst-package.Rd:54: Lost braces
54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
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checkRd: (-1) ddst-package.Rd:54: Lost braces
54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
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checkRd: (-1) ddst-package.Rd:58: Lost braces; missing escapes or markup?
58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
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58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
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checkRd: (-1) ddst-package.Rd:62: Lost braces; missing escapes or markup?
62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
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62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
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checkRd: (-1) ddst-package.Rd:65: Lost braces; missing escapes or markup?
65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
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checkRd: (-1) ddst-package.Rd:65: Lost braces; missing escapes or markup?
65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
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checkRd: (-1) ddst-package.Rd:67: Lost braces; missing escapes or markup?
67 | The choice of \emph{c} in \emph{T} and \emph{$T^*$} is decisive to finite sample behaviour of the selection rules and pertaining statistics \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$}. In particular, under large \emph{c}'s the rules behave similarly as Schwarz's (1978) BIC while for \emph{c=0} they mimic Akaike's (1973) AIC. For moderate sample sizes, values \emph{c in (2,2.5)} guarantee, under `smooth' departures, only slightly smaller power as in case BIC were used and simultaneously give much higher power than BIC under multimodal alternatives. In genral, large \emph{c's} are recommended if changes in location, scale, skewness and kurtosis are in principle aimed to be detected. For evidence and discussion see Inglot and Ledwina (2006 c).
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69 | It \emph{c>0} then the limiting null distribution of \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} is central chi-squared with one degree of freedom. In our implementation, for given \emph{n}, both critical values and \emph{p}-values are computed by MC method.
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71 | Empirical distributions of \emph{T} and \emph{$T^*$} as well as \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} are not essentially influenced by the choice of reasonably large \emph{d}'s, provided that sample size is at least moderate.
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checkRd: (-1) ddst.exp.test.Rd:27: Lost braces; missing escapes or markup?
27 | Modelling alternatives similarly as in Kallenberg and Ledwina (1997 a,b), e.g., and estimating \emph{$gamma$} by \emph{$tilde gamma= 1/n sum_{i=1}^n Z_i$} yields the efficient score
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checkRd: (-1) ddst.exp.test.Rd:30: Lost braces; missing escapes or markup?
30 | The matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and computed in a numerical way in case of cosine basis. In the implementation the default value of \emph{c} in \emph{$T^*$} is set to be 100.
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
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checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:30: Lost braces; missing escapes or markup?
30 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=1/n sum_{i=1}^n Z_i$} and
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checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
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31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
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32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
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25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
| ^
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25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
| ^
NOTE
r-devel-macos-arm64
Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:31: Lost braces; missing escapes or markup?
31 | where \emph{$l(Z_i)$}, i=1,...,n, is \emph{k}-dimensional (row) score vector, the symbol \emph{'} denotes transposition while \emph{$I=Cov_{theta_0}[l(Z_1)]'[l(Z_1)]$}. Following Neyman's idea of modelling underlying distributions one gets \emph{$l(Z_i)=(phi_1(F(Z_i)),...,phi_k(F(Z_i)))$} and \emph{I} being the identity matrix, where \emph{$phi_j$}'s, j >= 1, are zero mean orthonormal functions on [0,1], while \emph{F} is the completely specified null distribution function.
| ^
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35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
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35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:36: Lost braces; missing escapes or markup?
36 | where \emph{$tilde gamma$} is an appropriate estimator of \emph{$gamma$} while \emph{$I^*(gamma)=Cov_{theta_0}[l^*(Z_1;gamma)]'[l^*(Z_1;gamma)]$}. More details can be found in Janic and Ledwina (2008), Kallenberg and Ledwina (1997 a,b) as well as Inglot and Ledwina (2006 a,b).
| ^
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40 | \emph{$T = min{1 <= k <= d: W_k-pi(k,n,c) >= W_j-pi(j,n,c), j=1,...,d}$}
| ^
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45 | $T^* = min{1 <= k <= d: W_k^*(tilde gamma)-pi^*(k,n,c) >= W_j^*(tilde gamma)-pi^*(j,n,c), j=1,...,d}$}.
| ^
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49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
| ^
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49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
| ^
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49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
| ^
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54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
| ^
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54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
| ^
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58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
| ^
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58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
| ^
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62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
| ^
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62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
| ^
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65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
| ^
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65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
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67 | The choice of \emph{c} in \emph{T} and \emph{$T^*$} is decisive to finite sample behaviour of the selection rules and pertaining statistics \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$}. In particular, under large \emph{c}'s the rules behave similarly as Schwarz's (1978) BIC while for \emph{c=0} they mimic Akaike's (1973) AIC. For moderate sample sizes, values \emph{c in (2,2.5)} guarantee, under `smooth' departures, only slightly smaller power as in case BIC were used and simultaneously give much higher power than BIC under multimodal alternatives. In genral, large \emph{c's} are recommended if changes in location, scale, skewness and kurtosis are in principle aimed to be detected. For evidence and discussion see Inglot and Ledwina (2006 c).
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69 | It \emph{c>0} then the limiting null distribution of \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} is central chi-squared with one degree of freedom. In our implementation, for given \emph{n}, both critical values and \emph{p}-values are computed by MC method.
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71 | Empirical distributions of \emph{T} and \emph{$T^*$} as well as \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} are not essentially influenced by the choice of reasonably large \emph{d}'s, provided that sample size is at least moderate.
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27 | Modelling alternatives similarly as in Kallenberg and Ledwina (1997 a,b), e.g., and estimating \emph{$gamma$} by \emph{$tilde gamma= 1/n sum_{i=1}^n Z_i$} yields the efficient score
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30 | The matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and computed in a numerical way in case of cosine basis. In the implementation the default value of \emph{c} in \emph{$T^*$} is set to be 100.
| ^
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29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
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29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
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29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
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29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
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29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
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29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
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33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
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33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
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30 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=1/n sum_{i=1}^n Z_i$} and
| ^
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31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
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31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
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31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
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31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
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31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
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25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
| ^
checkRd: (-1) ddst.uniform.test.Rd:25: Lost braces; missing escapes or markup?
25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
| ^
NOTE
r-devel-windows-x86_64
Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
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29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
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29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
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31 | where \emph{$l(Z_i)$}, i=1,...,n, is \emph{k}-dimensional (row) score vector, the symbol \emph{'} denotes transposition while \emph{$I=Cov_{theta_0}[l(Z_1)]'[l(Z_1)]$}. Following Neyman's idea of modelling underlying distributions one gets \emph{$l(Z_i)=(phi_1(F(Z_i)),...,phi_k(F(Z_i)))$} and \emph{I} being the identity matrix, where \emph{$phi_j$}'s, j >= 1, are zero mean orthonormal functions on [0,1], while \emph{F} is the completely specified null distribution function.
| ^
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35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:36: Lost braces; missing escapes or markup?
36 | where \emph{$tilde gamma$} is an appropriate estimator of \emph{$gamma$} while \emph{$I^*(gamma)=Cov_{theta_0}[l^*(Z_1;gamma)]'[l^*(Z_1;gamma)]$}. More details can be found in Janic and Ledwina (2008), Kallenberg and Ledwina (1997 a,b) as well as Inglot and Ledwina (2006 a,b).
| ^
checkRd: (-1) ddst-package.Rd:40: Lost braces
40 | \emph{$T = min{1 <= k <= d: W_k-pi(k,n,c) >= W_j-pi(j,n,c), j=1,...,d}$}
| ^
checkRd: (-1) ddst-package.Rd:45: Lost braces
45 | $T^* = min{1 <= k <= d: W_k^*(tilde gamma)-pi^*(k,n,c) >= W_j^*(tilde gamma)-pi^*(j,n,c), j=1,...,d}$}.
| ^
checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
| ^
checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
| ^
checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
| ^
checkRd: (-1) ddst-package.Rd:54: Lost braces
54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
| ^
checkRd: (-1) ddst-package.Rd:54: Lost braces
54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
| ^
checkRd: (-1) ddst-package.Rd:58: Lost braces; missing escapes or markup?
58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
| ^
checkRd: (-1) ddst-package.Rd:58: Lost braces; missing escapes or markup?
58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
| ^
checkRd: (-1) ddst-package.Rd:62: Lost braces; missing escapes or markup?
62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
| ^
checkRd: (-1) ddst-package.Rd:62: Lost braces; missing escapes or markup?
62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
| ^
checkRd: (-1) ddst-package.Rd:65: Lost braces; missing escapes or markup?
65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
| ^
checkRd: (-1) ddst-package.Rd:65: Lost braces; missing escapes or markup?
65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
| ^
checkRd: (-1) ddst-package.Rd:67: Lost braces; missing escapes or markup?
67 | The choice of \emph{c} in \emph{T} and \emph{$T^*$} is decisive to finite sample behaviour of the selection rules and pertaining statistics \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$}. In particular, under large \emph{c}'s the rules behave similarly as Schwarz's (1978) BIC while for \emph{c=0} they mimic Akaike's (1973) AIC. For moderate sample sizes, values \emph{c in (2,2.5)} guarantee, under `smooth' departures, only slightly smaller power as in case BIC were used and simultaneously give much higher power than BIC under multimodal alternatives. In genral, large \emph{c's} are recommended if changes in location, scale, skewness and kurtosis are in principle aimed to be detected. For evidence and discussion see Inglot and Ledwina (2006 c).
| ^
checkRd: (-1) ddst-package.Rd:69: Lost braces; missing escapes or markup?
69 | It \emph{c>0} then the limiting null distribution of \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} is central chi-squared with one degree of freedom. In our implementation, for given \emph{n}, both critical values and \emph{p}-values are computed by MC method.
| ^
checkRd: (-1) ddst-package.Rd:71: Lost braces; missing escapes or markup?
71 | Empirical distributions of \emph{T} and \emph{$T^*$} as well as \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} are not essentially influenced by the choice of reasonably large \emph{d}'s, provided that sample size is at least moderate.
| ^
checkRd: (-1) ddst.exp.test.Rd:27: Lost braces; missing escapes or markup?
27 | Modelling alternatives similarly as in Kallenberg and Ledwina (1997 a,b), e.g., and estimating \emph{$gamma$} by \emph{$tilde gamma= 1/n sum_{i=1}^n Z_i$} yields the efficient score
| ^
checkRd: (-1) ddst.exp.test.Rd:30: Lost braces; missing escapes or markup?
30 | The matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and computed in a numerical way in case of cosine basis. In the implementation the default value of \emph{c} in \emph{$T^*$} is set to be 100.
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:30: Lost braces; missing escapes or markup?
30 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=1/n sum_{i=1}^n Z_i$} and
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.uniform.test.Rd:25: Lost braces; missing escapes or markup?
25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
| ^
checkRd: (-1) ddst.uniform.test.Rd:25: Lost braces; missing escapes or markup?
25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
| ^
NOTE
r-oldrel-macos-arm64
Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:31: Lost braces; missing escapes or markup?
31 | where \emph{$l(Z_i)$}, i=1,...,n, is \emph{k}-dimensional (row) score vector, the symbol \emph{'} denotes transposition while \emph{$I=Cov_{theta_0}[l(Z_1)]'[l(Z_1)]$}. Following Neyman's idea of modelling underlying distributions one gets \emph{$l(Z_i)=(phi_1(F(Z_i)),...,phi_k(F(Z_i)))$} and \emph{I} being the identity matrix, where \emph{$phi_j$}'s, j >= 1, are zero mean orthonormal functions on [0,1], while \emph{F} is the completely specified null distribution function.
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:36: Lost braces; missing escapes or markup?
36 | where \emph{$tilde gamma$} is an appropriate estimator of \emph{$gamma$} while \emph{$I^*(gamma)=Cov_{theta_0}[l^*(Z_1;gamma)]'[l^*(Z_1;gamma)]$}. More details can be found in Janic and Ledwina (2008), Kallenberg and Ledwina (1997 a,b) as well as Inglot and Ledwina (2006 a,b).
| ^
checkRd: (-1) ddst-package.Rd:40: Lost braces
40 | \emph{$T = min{1 <= k <= d: W_k-pi(k,n,c) >= W_j-pi(j,n,c), j=1,...,d}$}
| ^
checkRd: (-1) ddst-package.Rd:45: Lost braces
45 | $T^* = min{1 <= k <= d: W_k^*(tilde gamma)-pi^*(k,n,c) >= W_j^*(tilde gamma)-pi^*(j,n,c), j=1,...,d}$}.
| ^
checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
| ^
checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
| ^
checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
| ^
checkRd: (-1) ddst-package.Rd:54: Lost braces
54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
| ^
checkRd: (-1) ddst-package.Rd:54: Lost braces
54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
| ^
checkRd: (-1) ddst-package.Rd:58: Lost braces; missing escapes or markup?
58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
| ^
checkRd: (-1) ddst-package.Rd:58: Lost braces; missing escapes or markup?
58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
| ^
checkRd: (-1) ddst-package.Rd:62: Lost braces; missing escapes or markup?
62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
| ^
checkRd: (-1) ddst-package.Rd:62: Lost braces; missing escapes or markup?
62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
| ^
checkRd: (-1) ddst-package.Rd:65: Lost braces; missing escapes or markup?
65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
| ^
checkRd: (-1) ddst-package.Rd:65: Lost braces; missing escapes or markup?
65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
| ^
checkRd: (-1) ddst-package.Rd:67: Lost braces; missing escapes or markup?
67 | The choice of \emph{c} in \emph{T} and \emph{$T^*$} is decisive to finite sample behaviour of the selection rules and pertaining statistics \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$}. In particular, under large \emph{c}'s the rules behave similarly as Schwarz's (1978) BIC while for \emph{c=0} they mimic Akaike's (1973) AIC. For moderate sample sizes, values \emph{c in (2,2.5)} guarantee, under `smooth' departures, only slightly smaller power as in case BIC were used and simultaneously give much higher power than BIC under multimodal alternatives. In genral, large \emph{c's} are recommended if changes in location, scale, skewness and kurtosis are in principle aimed to be detected. For evidence and discussion see Inglot and Ledwina (2006 c).
| ^
checkRd: (-1) ddst-package.Rd:69: Lost braces; missing escapes or markup?
69 | It \emph{c>0} then the limiting null distribution of \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} is central chi-squared with one degree of freedom. In our implementation, for given \emph{n}, both critical values and \emph{p}-values are computed by MC method.
| ^
checkRd: (-1) ddst-package.Rd:71: Lost braces; missing escapes or markup?
71 | Empirical distributions of \emph{T} and \emph{$T^*$} as well as \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} are not essentially influenced by the choice of reasonably large \emph{d}'s, provided that sample size is at least moderate.
| ^
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27 | Modelling alternatives similarly as in Kallenberg and Ledwina (1997 a,b), e.g., and estimating \emph{$gamma$} by \emph{$tilde gamma= 1/n sum_{i=1}^n Z_i$} yields the efficient score
| ^
checkRd: (-1) ddst.exp.test.Rd:30: Lost braces; missing escapes or markup?
30 | The matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and computed in a numerical way in case of cosine basis. In the implementation the default value of \emph{c} in \emph{$T^*$} is set to be 100.
| ^
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29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:30: Lost braces; missing escapes or markup?
30 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=1/n sum_{i=1}^n Z_i$} and
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
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checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
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checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
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checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
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checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
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checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
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checkRd: (-1) ddst.uniform.test.Rd:25: Lost braces; missing escapes or markup?
25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
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25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
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NOTE
r-oldrel-macos-x86_64
Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
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29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
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29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
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checkRd: (-1) ddst-package.Rd:31: Lost braces; missing escapes or markup?
31 | where \emph{$l(Z_i)$}, i=1,...,n, is \emph{k}-dimensional (row) score vector, the symbol \emph{'} denotes transposition while \emph{$I=Cov_{theta_0}[l(Z_1)]'[l(Z_1)]$}. Following Neyman's idea of modelling underlying distributions one gets \emph{$l(Z_i)=(phi_1(F(Z_i)),...,phi_k(F(Z_i)))$} and \emph{I} being the identity matrix, where \emph{$phi_j$}'s, j >= 1, are zero mean orthonormal functions on [0,1], while \emph{F} is the completely specified null distribution function.
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35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
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35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
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35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
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35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
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checkRd: (-1) ddst-package.Rd:36: Lost braces; missing escapes or markup?
36 | where \emph{$tilde gamma$} is an appropriate estimator of \emph{$gamma$} while \emph{$I^*(gamma)=Cov_{theta_0}[l^*(Z_1;gamma)]'[l^*(Z_1;gamma)]$}. More details can be found in Janic and Ledwina (2008), Kallenberg and Ledwina (1997 a,b) as well as Inglot and Ledwina (2006 a,b).
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40 | \emph{$T = min{1 <= k <= d: W_k-pi(k,n,c) >= W_j-pi(j,n,c), j=1,...,d}$}
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45 | $T^* = min{1 <= k <= d: W_k^*(tilde gamma)-pi^*(k,n,c) >= W_j^*(tilde gamma)-pi^*(j,n,c), j=1,...,d}$}.
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checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
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checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
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checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
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checkRd: (-1) ddst-package.Rd:54: Lost braces
54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
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checkRd: (-1) ddst-package.Rd:54: Lost braces
54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
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checkRd: (-1) ddst-package.Rd:58: Lost braces; missing escapes or markup?
58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
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58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
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checkRd: (-1) ddst-package.Rd:62: Lost braces; missing escapes or markup?
62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
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62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
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checkRd: (-1) ddst-package.Rd:65: Lost braces; missing escapes or markup?
65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
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checkRd: (-1) ddst-package.Rd:65: Lost braces; missing escapes or markup?
65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
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checkRd: (-1) ddst-package.Rd:67: Lost braces; missing escapes or markup?
67 | The choice of \emph{c} in \emph{T} and \emph{$T^*$} is decisive to finite sample behaviour of the selection rules and pertaining statistics \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$}. In particular, under large \emph{c}'s the rules behave similarly as Schwarz's (1978) BIC while for \emph{c=0} they mimic Akaike's (1973) AIC. For moderate sample sizes, values \emph{c in (2,2.5)} guarantee, under `smooth' departures, only slightly smaller power as in case BIC were used and simultaneously give much higher power than BIC under multimodal alternatives. In genral, large \emph{c's} are recommended if changes in location, scale, skewness and kurtosis are in principle aimed to be detected. For evidence and discussion see Inglot and Ledwina (2006 c).
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69 | It \emph{c>0} then the limiting null distribution of \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} is central chi-squared with one degree of freedom. In our implementation, for given \emph{n}, both critical values and \emph{p}-values are computed by MC method.
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71 | Empirical distributions of \emph{T} and \emph{$T^*$} as well as \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} are not essentially influenced by the choice of reasonably large \emph{d}'s, provided that sample size is at least moderate.
| ^
checkRd: (-1) ddst.exp.test.Rd:27: Lost braces; missing escapes or markup?
27 | Modelling alternatives similarly as in Kallenberg and Ledwina (1997 a,b), e.g., and estimating \emph{$gamma$} by \emph{$tilde gamma= 1/n sum_{i=1}^n Z_i$} yields the efficient score
| ^
checkRd: (-1) ddst.exp.test.Rd:30: Lost braces; missing escapes or markup?
30 | The matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and computed in a numerical way in case of cosine basis. In the implementation the default value of \emph{c} in \emph{$T^*$} is set to be 100.
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
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checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
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checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
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checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
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checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
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checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
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33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
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30 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=1/n sum_{i=1}^n Z_i$} and
| ^
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31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
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31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
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31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
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31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
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31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
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32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
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32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
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32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
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25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
| ^
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25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
| ^
NOTE
r-oldrel-windows-x86_64
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29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
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29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
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29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
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31 | where \emph{$l(Z_i)$}, i=1,...,n, is \emph{k}-dimensional (row) score vector, the symbol \emph{'} denotes transposition while \emph{$I=Cov_{theta_0}[l(Z_1)]'[l(Z_1)]$}. Following Neyman's idea of modelling underlying distributions one gets \emph{$l(Z_i)=(phi_1(F(Z_i)),...,phi_k(F(Z_i)))$} and \emph{I} being the identity matrix, where \emph{$phi_j$}'s, j >= 1, are zero mean orthonormal functions on [0,1], while \emph{F} is the completely specified null distribution function.
| ^
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35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
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35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
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35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
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35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
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36 | where \emph{$tilde gamma$} is an appropriate estimator of \emph{$gamma$} while \emph{$I^*(gamma)=Cov_{theta_0}[l^*(Z_1;gamma)]'[l^*(Z_1;gamma)]$}. More details can be found in Janic and Ledwina (2008), Kallenberg and Ledwina (1997 a,b) as well as Inglot and Ledwina (2006 a,b).
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40 | \emph{$T = min{1 <= k <= d: W_k-pi(k,n,c) >= W_j-pi(j,n,c), j=1,...,d}$}
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45 | $T^* = min{1 <= k <= d: W_k^*(tilde gamma)-pi^*(k,n,c) >= W_j^*(tilde gamma)-pi^*(j,n,c), j=1,...,d}$}.
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49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
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49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
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49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
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54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
| ^
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54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
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58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
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58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
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62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
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62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
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65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
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65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
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67 | The choice of \emph{c} in \emph{T} and \emph{$T^*$} is decisive to finite sample behaviour of the selection rules and pertaining statistics \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$}. In particular, under large \emph{c}'s the rules behave similarly as Schwarz's (1978) BIC while for \emph{c=0} they mimic Akaike's (1973) AIC. For moderate sample sizes, values \emph{c in (2,2.5)} guarantee, under `smooth' departures, only slightly smaller power as in case BIC were used and simultaneously give much higher power than BIC under multimodal alternatives. In genral, large \emph{c's} are recommended if changes in location, scale, skewness and kurtosis are in principle aimed to be detected. For evidence and discussion see Inglot and Ledwina (2006 c).
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69 | It \emph{c>0} then the limiting null distribution of \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} is central chi-squared with one degree of freedom. In our implementation, for given \emph{n}, both critical values and \emph{p}-values are computed by MC method.
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71 | Empirical distributions of \emph{T} and \emph{$T^*$} as well as \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} are not essentially influenced by the choice of reasonably large \emph{d}'s, provided that sample size is at least moderate.
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27 | Modelling alternatives similarly as in Kallenberg and Ledwina (1997 a,b), e.g., and estimating \emph{$gamma$} by \emph{$tilde gamma= 1/n sum_{i=1}^n Z_i$} yields the efficient score
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30 | The matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and computed in a numerical way in case of cosine basis. In the implementation the default value of \emph{c} in \emph{$T^*$} is set to be 100.
| ^
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29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
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29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
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29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
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29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
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29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
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29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
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29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
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33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
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33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
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33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
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30 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=1/n sum_{i=1}^n Z_i$} and
| ^
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31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
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31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
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32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
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32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
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35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
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35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
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35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
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25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
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25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
| ^
NOTE
r-patched-linux-x86_64
Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
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checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
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checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
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checkRd: (-1) ddst-package.Rd:31: Lost braces; missing escapes or markup?
31 | where \emph{$l(Z_i)$}, i=1,...,n, is \emph{k}-dimensional (row) score vector, the symbol \emph{'} denotes transposition while \emph{$I=Cov_{theta_0}[l(Z_1)]'[l(Z_1)]$}. Following Neyman's idea of modelling underlying distributions one gets \emph{$l(Z_i)=(phi_1(F(Z_i)),...,phi_k(F(Z_i)))$} and \emph{I} being the identity matrix, where \emph{$phi_j$}'s, j >= 1, are zero mean orthonormal functions on [0,1], while \emph{F} is the completely specified null distribution function.
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checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
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checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
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checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
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checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
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checkRd: (-1) ddst-package.Rd:36: Lost braces; missing escapes or markup?
36 | where \emph{$tilde gamma$} is an appropriate estimator of \emph{$gamma$} while \emph{$I^*(gamma)=Cov_{theta_0}[l^*(Z_1;gamma)]'[l^*(Z_1;gamma)]$}. More details can be found in Janic and Ledwina (2008), Kallenberg and Ledwina (1997 a,b) as well as Inglot and Ledwina (2006 a,b).
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checkRd: (-1) ddst-package.Rd:40: Lost braces
40 | \emph{$T = min{1 <= k <= d: W_k-pi(k,n,c) >= W_j-pi(j,n,c), j=1,...,d}$}
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checkRd: (-1) ddst-package.Rd:45: Lost braces
45 | $T^* = min{1 <= k <= d: W_k^*(tilde gamma)-pi^*(k,n,c) >= W_j^*(tilde gamma)-pi^*(j,n,c), j=1,...,d}$}.
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checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
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checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
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checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
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checkRd: (-1) ddst-package.Rd:54: Lost braces
54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
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checkRd: (-1) ddst-package.Rd:54: Lost braces
54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
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checkRd: (-1) ddst-package.Rd:58: Lost braces; missing escapes or markup?
58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
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checkRd: (-1) ddst-package.Rd:58: Lost braces; missing escapes or markup?
58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
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checkRd: (-1) ddst-package.Rd:62: Lost braces; missing escapes or markup?
62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
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checkRd: (-1) ddst-package.Rd:62: Lost braces; missing escapes or markup?
62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
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checkRd: (-1) ddst-package.Rd:65: Lost braces; missing escapes or markup?
65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
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checkRd: (-1) ddst-package.Rd:65: Lost braces; missing escapes or markup?
65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
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checkRd: (-1) ddst-package.Rd:67: Lost braces; missing escapes or markup?
67 | The choice of \emph{c} in \emph{T} and \emph{$T^*$} is decisive to finite sample behaviour of the selection rules and pertaining statistics \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$}. In particular, under large \emph{c}'s the rules behave similarly as Schwarz's (1978) BIC while for \emph{c=0} they mimic Akaike's (1973) AIC. For moderate sample sizes, values \emph{c in (2,2.5)} guarantee, under `smooth' departures, only slightly smaller power as in case BIC were used and simultaneously give much higher power than BIC under multimodal alternatives. In genral, large \emph{c's} are recommended if changes in location, scale, skewness and kurtosis are in principle aimed to be detected. For evidence and discussion see Inglot and Ledwina (2006 c).
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checkRd: (-1) ddst-package.Rd:69: Lost braces; missing escapes or markup?
69 | It \emph{c>0} then the limiting null distribution of \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} is central chi-squared with one degree of freedom. In our implementation, for given \emph{n}, both critical values and \emph{p}-values are computed by MC method.
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checkRd: (-1) ddst-package.Rd:71: Lost braces; missing escapes or markup?
71 | Empirical distributions of \emph{T} and \emph{$T^*$} as well as \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} are not essentially influenced by the choice of reasonably large \emph{d}'s, provided that sample size is at least moderate.
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checkRd: (-1) ddst.exp.test.Rd:27: Lost braces; missing escapes or markup?
27 | Modelling alternatives similarly as in Kallenberg and Ledwina (1997 a,b), e.g., and estimating \emph{$gamma$} by \emph{$tilde gamma= 1/n sum_{i=1}^n Z_i$} yields the efficient score
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checkRd: (-1) ddst.exp.test.Rd:30: Lost braces; missing escapes or markup?
30 | The matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and computed in a numerical way in case of cosine basis. In the implementation the default value of \emph{c} in \emph{$T^*$} is set to be 100.
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checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
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checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
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checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
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checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
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checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
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checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
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checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
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checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
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checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
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checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
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checkRd: (-1) ddst.norm.test.Rd:30: Lost braces; missing escapes or markup?
30 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=1/n sum_{i=1}^n Z_i$} and
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checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
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checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
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checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
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checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
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checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
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checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
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checkRd: (-1) ddst.uniform.test.Rd:25: Lost braces; missing escapes or markup?
25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
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checkRd: (-1) ddst.uniform.test.Rd:25: Lost braces; missing escapes or markup?
25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
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NOTE
r-release-linux-x86_64
Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
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checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
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checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
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checkRd: (-1) ddst-package.Rd:31: Lost braces; missing escapes or markup?
31 | where \emph{$l(Z_i)$}, i=1,...,n, is \emph{k}-dimensional (row) score vector, the symbol \emph{'} denotes transposition while \emph{$I=Cov_{theta_0}[l(Z_1)]'[l(Z_1)]$}. Following Neyman's idea of modelling underlying distributions one gets \emph{$l(Z_i)=(phi_1(F(Z_i)),...,phi_k(F(Z_i)))$} and \emph{I} being the identity matrix, where \emph{$phi_j$}'s, j >= 1, are zero mean orthonormal functions on [0,1], while \emph{F} is the completely specified null distribution function.
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checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
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checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
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checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
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checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
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checkRd: (-1) ddst-package.Rd:36: Lost braces; missing escapes or markup?
36 | where \emph{$tilde gamma$} is an appropriate estimator of \emph{$gamma$} while \emph{$I^*(gamma)=Cov_{theta_0}[l^*(Z_1;gamma)]'[l^*(Z_1;gamma)]$}. More details can be found in Janic and Ledwina (2008), Kallenberg and Ledwina (1997 a,b) as well as Inglot and Ledwina (2006 a,b).
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checkRd: (-1) ddst-package.Rd:40: Lost braces
40 | \emph{$T = min{1 <= k <= d: W_k-pi(k,n,c) >= W_j-pi(j,n,c), j=1,...,d}$}
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checkRd: (-1) ddst-package.Rd:45: Lost braces
45 | $T^* = min{1 <= k <= d: W_k^*(tilde gamma)-pi^*(k,n,c) >= W_j^*(tilde gamma)-pi^*(j,n,c), j=1,...,d}$}.
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checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
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checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
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checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
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checkRd: (-1) ddst-package.Rd:54: Lost braces
54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
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checkRd: (-1) ddst-package.Rd:54: Lost braces
54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
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checkRd: (-1) ddst-package.Rd:58: Lost braces; missing escapes or markup?
58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
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checkRd: (-1) ddst-package.Rd:58: Lost braces; missing escapes or markup?
58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
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checkRd: (-1) ddst-package.Rd:62: Lost braces; missing escapes or markup?
62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
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checkRd: (-1) ddst-package.Rd:62: Lost braces; missing escapes or markup?
62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
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checkRd: (-1) ddst-package.Rd:65: Lost braces; missing escapes or markup?
65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
| ^
checkRd: (-1) ddst-package.Rd:65: Lost braces; missing escapes or markup?
65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
| ^
checkRd: (-1) ddst-package.Rd:67: Lost braces; missing escapes or markup?
67 | The choice of \emph{c} in \emph{T} and \emph{$T^*$} is decisive to finite sample behaviour of the selection rules and pertaining statistics \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$}. In particular, under large \emph{c}'s the rules behave similarly as Schwarz's (1978) BIC while for \emph{c=0} they mimic Akaike's (1973) AIC. For moderate sample sizes, values \emph{c in (2,2.5)} guarantee, under `smooth' departures, only slightly smaller power as in case BIC were used and simultaneously give much higher power than BIC under multimodal alternatives. In genral, large \emph{c's} are recommended if changes in location, scale, skewness and kurtosis are in principle aimed to be detected. For evidence and discussion see Inglot and Ledwina (2006 c).
| ^
checkRd: (-1) ddst-package.Rd:69: Lost braces; missing escapes or markup?
69 | It \emph{c>0} then the limiting null distribution of \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} is central chi-squared with one degree of freedom. In our implementation, for given \emph{n}, both critical values and \emph{p}-values are computed by MC method.
| ^
checkRd: (-1) ddst-package.Rd:71: Lost braces; missing escapes or markup?
71 | Empirical distributions of \emph{T} and \emph{$T^*$} as well as \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} are not essentially influenced by the choice of reasonably large \emph{d}'s, provided that sample size is at least moderate.
| ^
checkRd: (-1) ddst.exp.test.Rd:27: Lost braces; missing escapes or markup?
27 | Modelling alternatives similarly as in Kallenberg and Ledwina (1997 a,b), e.g., and estimating \emph{$gamma$} by \emph{$tilde gamma= 1/n sum_{i=1}^n Z_i$} yields the efficient score
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checkRd: (-1) ddst.exp.test.Rd:30: Lost braces; missing escapes or markup?
30 | The matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and computed in a numerical way in case of cosine basis. In the implementation the default value of \emph{c} in \emph{$T^*$} is set to be 100.
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
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checkRd: (-1) ddst.norm.test.Rd:30: Lost braces; missing escapes or markup?
30 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=1/n sum_{i=1}^n Z_i$} and
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checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
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checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
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checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
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checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
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checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
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checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
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checkRd: (-1) ddst.uniform.test.Rd:25: Lost braces; missing escapes or markup?
25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
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checkRd: (-1) ddst.uniform.test.Rd:25: Lost braces; missing escapes or markup?
25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
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NOTE
r-release-macos-arm64
Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
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checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
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checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
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checkRd: (-1) ddst-package.Rd:31: Lost braces; missing escapes or markup?
31 | where \emph{$l(Z_i)$}, i=1,...,n, is \emph{k}-dimensional (row) score vector, the symbol \emph{'} denotes transposition while \emph{$I=Cov_{theta_0}[l(Z_1)]'[l(Z_1)]$}. Following Neyman's idea of modelling underlying distributions one gets \emph{$l(Z_i)=(phi_1(F(Z_i)),...,phi_k(F(Z_i)))$} and \emph{I} being the identity matrix, where \emph{$phi_j$}'s, j >= 1, are zero mean orthonormal functions on [0,1], while \emph{F} is the completely specified null distribution function.
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checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
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checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
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checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
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checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
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checkRd: (-1) ddst-package.Rd:36: Lost braces; missing escapes or markup?
36 | where \emph{$tilde gamma$} is an appropriate estimator of \emph{$gamma$} while \emph{$I^*(gamma)=Cov_{theta_0}[l^*(Z_1;gamma)]'[l^*(Z_1;gamma)]$}. More details can be found in Janic and Ledwina (2008), Kallenberg and Ledwina (1997 a,b) as well as Inglot and Ledwina (2006 a,b).
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checkRd: (-1) ddst-package.Rd:40: Lost braces
40 | \emph{$T = min{1 <= k <= d: W_k-pi(k,n,c) >= W_j-pi(j,n,c), j=1,...,d}$}
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checkRd: (-1) ddst-package.Rd:45: Lost braces
45 | $T^* = min{1 <= k <= d: W_k^*(tilde gamma)-pi^*(k,n,c) >= W_j^*(tilde gamma)-pi^*(j,n,c), j=1,...,d}$}.
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checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
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checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
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checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
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checkRd: (-1) ddst-package.Rd:54: Lost braces
54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
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checkRd: (-1) ddst-package.Rd:54: Lost braces
54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
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checkRd: (-1) ddst-package.Rd:58: Lost braces; missing escapes or markup?
58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
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checkRd: (-1) ddst-package.Rd:58: Lost braces; missing escapes or markup?
58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
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checkRd: (-1) ddst-package.Rd:62: Lost braces; missing escapes or markup?
62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
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checkRd: (-1) ddst-package.Rd:62: Lost braces; missing escapes or markup?
62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
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checkRd: (-1) ddst-package.Rd:65: Lost braces; missing escapes or markup?
65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
| ^
checkRd: (-1) ddst-package.Rd:65: Lost braces; missing escapes or markup?
65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
| ^
checkRd: (-1) ddst-package.Rd:67: Lost braces; missing escapes or markup?
67 | The choice of \emph{c} in \emph{T} and \emph{$T^*$} is decisive to finite sample behaviour of the selection rules and pertaining statistics \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$}. In particular, under large \emph{c}'s the rules behave similarly as Schwarz's (1978) BIC while for \emph{c=0} they mimic Akaike's (1973) AIC. For moderate sample sizes, values \emph{c in (2,2.5)} guarantee, under `smooth' departures, only slightly smaller power as in case BIC were used and simultaneously give much higher power than BIC under multimodal alternatives. In genral, large \emph{c's} are recommended if changes in location, scale, skewness and kurtosis are in principle aimed to be detected. For evidence and discussion see Inglot and Ledwina (2006 c).
| ^
checkRd: (-1) ddst-package.Rd:69: Lost braces; missing escapes or markup?
69 | It \emph{c>0} then the limiting null distribution of \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} is central chi-squared with one degree of freedom. In our implementation, for given \emph{n}, both critical values and \emph{p}-values are computed by MC method.
| ^
checkRd: (-1) ddst-package.Rd:71: Lost braces; missing escapes or markup?
71 | Empirical distributions of \emph{T} and \emph{$T^*$} as well as \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} are not essentially influenced by the choice of reasonably large \emph{d}'s, provided that sample size is at least moderate.
| ^
checkRd: (-1) ddst.exp.test.Rd:27: Lost braces; missing escapes or markup?
27 | Modelling alternatives similarly as in Kallenberg and Ledwina (1997 a,b), e.g., and estimating \emph{$gamma$} by \emph{$tilde gamma= 1/n sum_{i=1}^n Z_i$} yields the efficient score
| ^
checkRd: (-1) ddst.exp.test.Rd:30: Lost braces; missing escapes or markup?
30 | The matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and computed in a numerical way in case of cosine basis. In the implementation the default value of \emph{c} in \emph{$T^*$} is set to be 100.
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
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checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:30: Lost braces; missing escapes or markup?
30 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=1/n sum_{i=1}^n Z_i$} and
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.uniform.test.Rd:25: Lost braces; missing escapes or markup?
25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
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checkRd: (-1) ddst.uniform.test.Rd:25: Lost braces; missing escapes or markup?
25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
| ^
NOTE
r-release-macos-x86_64
Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
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checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
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checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
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checkRd: (-1) ddst-package.Rd:31: Lost braces; missing escapes or markup?
31 | where \emph{$l(Z_i)$}, i=1,...,n, is \emph{k}-dimensional (row) score vector, the symbol \emph{'} denotes transposition while \emph{$I=Cov_{theta_0}[l(Z_1)]'[l(Z_1)]$}. Following Neyman's idea of modelling underlying distributions one gets \emph{$l(Z_i)=(phi_1(F(Z_i)),...,phi_k(F(Z_i)))$} and \emph{I} being the identity matrix, where \emph{$phi_j$}'s, j >= 1, are zero mean orthonormal functions on [0,1], while \emph{F} is the completely specified null distribution function.
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
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checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:36: Lost braces; missing escapes or markup?
36 | where \emph{$tilde gamma$} is an appropriate estimator of \emph{$gamma$} while \emph{$I^*(gamma)=Cov_{theta_0}[l^*(Z_1;gamma)]'[l^*(Z_1;gamma)]$}. More details can be found in Janic and Ledwina (2008), Kallenberg and Ledwina (1997 a,b) as well as Inglot and Ledwina (2006 a,b).
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checkRd: (-1) ddst-package.Rd:40: Lost braces
40 | \emph{$T = min{1 <= k <= d: W_k-pi(k,n,c) >= W_j-pi(j,n,c), j=1,...,d}$}
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45 | $T^* = min{1 <= k <= d: W_k^*(tilde gamma)-pi^*(k,n,c) >= W_j^*(tilde gamma)-pi^*(j,n,c), j=1,...,d}$}.
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checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
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checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
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checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
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checkRd: (-1) ddst-package.Rd:54: Lost braces
54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
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checkRd: (-1) ddst-package.Rd:54: Lost braces
54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
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checkRd: (-1) ddst-package.Rd:58: Lost braces; missing escapes or markup?
58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
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checkRd: (-1) ddst-package.Rd:58: Lost braces; missing escapes or markup?
58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
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checkRd: (-1) ddst-package.Rd:62: Lost braces; missing escapes or markup?
62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
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checkRd: (-1) ddst-package.Rd:62: Lost braces; missing escapes or markup?
62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
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checkRd: (-1) ddst-package.Rd:65: Lost braces; missing escapes or markup?
65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
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checkRd: (-1) ddst-package.Rd:65: Lost braces; missing escapes or markup?
65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
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checkRd: (-1) ddst-package.Rd:67: Lost braces; missing escapes or markup?
67 | The choice of \emph{c} in \emph{T} and \emph{$T^*$} is decisive to finite sample behaviour of the selection rules and pertaining statistics \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$}. In particular, under large \emph{c}'s the rules behave similarly as Schwarz's (1978) BIC while for \emph{c=0} they mimic Akaike's (1973) AIC. For moderate sample sizes, values \emph{c in (2,2.5)} guarantee, under `smooth' departures, only slightly smaller power as in case BIC were used and simultaneously give much higher power than BIC under multimodal alternatives. In genral, large \emph{c's} are recommended if changes in location, scale, skewness and kurtosis are in principle aimed to be detected. For evidence and discussion see Inglot and Ledwina (2006 c).
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69 | It \emph{c>0} then the limiting null distribution of \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} is central chi-squared with one degree of freedom. In our implementation, for given \emph{n}, both critical values and \emph{p}-values are computed by MC method.
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71 | Empirical distributions of \emph{T} and \emph{$T^*$} as well as \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} are not essentially influenced by the choice of reasonably large \emph{d}'s, provided that sample size is at least moderate.
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checkRd: (-1) ddst.exp.test.Rd:27: Lost braces; missing escapes or markup?
27 | Modelling alternatives similarly as in Kallenberg and Ledwina (1997 a,b), e.g., and estimating \emph{$gamma$} by \emph{$tilde gamma= 1/n sum_{i=1}^n Z_i$} yields the efficient score
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checkRd: (-1) ddst.exp.test.Rd:30: Lost braces; missing escapes or markup?
30 | The matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and computed in a numerical way in case of cosine basis. In the implementation the default value of \emph{c} in \emph{$T^*$} is set to be 100.
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:30: Lost braces; missing escapes or markup?
30 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=1/n sum_{i=1}^n Z_i$} and
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
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checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.uniform.test.Rd:25: Lost braces; missing escapes or markup?
25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
| ^
checkRd: (-1) ddst.uniform.test.Rd:25: Lost braces; missing escapes or markup?
25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
| ^
NOTE
r-release-windows-x86_64
Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:31: Lost braces; missing escapes or markup?
31 | where \emph{$l(Z_i)$}, i=1,...,n, is \emph{k}-dimensional (row) score vector, the symbol \emph{'} denotes transposition while \emph{$I=Cov_{theta_0}[l(Z_1)]'[l(Z_1)]$}. Following Neyman's idea of modelling underlying distributions one gets \emph{$l(Z_i)=(phi_1(F(Z_i)),...,phi_k(F(Z_i)))$} and \emph{I} being the identity matrix, where \emph{$phi_j$}'s, j >= 1, are zero mean orthonormal functions on [0,1], while \emph{F} is the completely specified null distribution function.
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:35: Lost braces; missing escapes or markup?
35 | \emph{$W_k^{*}(tilde gamma)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)][I^*(tilde gamma)]^{-1}[1/sqrt(n) sum_{i=1}^n l^*(Z_i;tilde gamma)]'$},
| ^
checkRd: (-1) ddst-package.Rd:36: Lost braces; missing escapes or markup?
36 | where \emph{$tilde gamma$} is an appropriate estimator of \emph{$gamma$} while \emph{$I^*(gamma)=Cov_{theta_0}[l^*(Z_1;gamma)]'[l^*(Z_1;gamma)]$}. More details can be found in Janic and Ledwina (2008), Kallenberg and Ledwina (1997 a,b) as well as Inglot and Ledwina (2006 a,b).
| ^
checkRd: (-1) ddst-package.Rd:40: Lost braces
40 | \emph{$T = min{1 <= k <= d: W_k-pi(k,n,c) >= W_j-pi(j,n,c), j=1,...,d}$}
| ^
checkRd: (-1) ddst-package.Rd:45: Lost braces
45 | $T^* = min{1 <= k <= d: W_k^*(tilde gamma)-pi^*(k,n,c) >= W_j^*(tilde gamma)-pi^*(j,n,c), j=1,...,d}$}.
| ^
checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
| ^
checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
| ^
checkRd: (-1) ddst-package.Rd:49: Lost braces
49 | \emph{$pi(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k| <= sqrt(c log(n)), 2j, if max{1 <= k <= d}|Y_k|>sqrt(c log(n)). }$}
| ^
checkRd: (-1) ddst-package.Rd:54: Lost braces
54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
| ^
checkRd: (-1) ddst-package.Rd:54: Lost braces
54 | $pi^*(j,n,c)={jlog n, if max{1 <= k <= d}|Y_k^*| <= sqrt(c log(n)),2j if max(1 <= k <= d)|Y_k^*| > sqrt(c log(n))}$}.
| ^
checkRd: (-1) ddst-package.Rd:58: Lost braces; missing escapes or markup?
58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
| ^
checkRd: (-1) ddst-package.Rd:58: Lost braces; missing escapes or markup?
58 | \emph{$(Y_1,...,Y_k)=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1/2}$}
| ^
checkRd: (-1) ddst-package.Rd:62: Lost braces; missing escapes or markup?
62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
| ^
checkRd: (-1) ddst-package.Rd:62: Lost braces; missing escapes or markup?
62 | \emph{$(Y_1^*,...,Y_k^*)=[1/sqrt(n) sum_{i=1}^n l^*(Z_i; tilde gamma)][I^*(tilde gamma)]^{-1/2}$}.
| ^
checkRd: (-1) ddst-package.Rd:65: Lost braces; missing escapes or markup?
65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
| ^
checkRd: (-1) ddst-package.Rd:65: Lost braces; missing escapes or markup?
65 | and \emph{$W_{T^*} = W_{T^*}(tilde gamma)$}, respectively. For details see Inglot and Ledwina (2006 a,b,c).
| ^
checkRd: (-1) ddst-package.Rd:67: Lost braces; missing escapes or markup?
67 | The choice of \emph{c} in \emph{T} and \emph{$T^*$} is decisive to finite sample behaviour of the selection rules and pertaining statistics \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$}. In particular, under large \emph{c}'s the rules behave similarly as Schwarz's (1978) BIC while for \emph{c=0} they mimic Akaike's (1973) AIC. For moderate sample sizes, values \emph{c in (2,2.5)} guarantee, under `smooth' departures, only slightly smaller power as in case BIC were used and simultaneously give much higher power than BIC under multimodal alternatives. In genral, large \emph{c's} are recommended if changes in location, scale, skewness and kurtosis are in principle aimed to be detected. For evidence and discussion see Inglot and Ledwina (2006 c).
| ^
checkRd: (-1) ddst-package.Rd:69: Lost braces; missing escapes or markup?
69 | It \emph{c>0} then the limiting null distribution of \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} is central chi-squared with one degree of freedom. In our implementation, for given \emph{n}, both critical values and \emph{p}-values are computed by MC method.
| ^
checkRd: (-1) ddst-package.Rd:71: Lost braces; missing escapes or markup?
71 | Empirical distributions of \emph{T} and \emph{$T^*$} as well as \emph{$W_T$} and \emph{$W_{T^*}(tilde gamma)$} are not essentially influenced by the choice of reasonably large \emph{d}'s, provided that sample size is at least moderate.
| ^
checkRd: (-1) ddst.exp.test.Rd:27: Lost braces; missing escapes or markup?
27 | Modelling alternatives similarly as in Kallenberg and Ledwina (1997 a,b), e.g., and estimating \emph{$gamma$} by \emph{$tilde gamma= 1/n sum_{i=1}^n Z_i$} yields the efficient score
| ^
checkRd: (-1) ddst.exp.test.Rd:30: Lost braces; missing escapes or markup?
30 | The matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and computed in a numerical way in case of cosine basis. In the implementation the default value of \emph{c} in \emph{$T^*$} is set to be 100.
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=-1/n sum_{i=1}^n Z_i + varepsilon G$}, where \emph{$varepsilon approx 0.577216 $} is the Euler constant and \emph{$ G = tilde gamma_2 = [n(n-1) ln2]^{-1}sum_{1<= j < i <= n}(Z_{n:i}^o - Z_{n:j}^o) $} while \emph{$Z_{n:1}^o <= ... <= Z_{n:n}^o$}
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.extr.test.Rd:33: Lost braces; missing escapes or markup?
33 | The related matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of \emph{c} in \emph{$T^*$} was fixed to be 100. Hence, \emph{$T^*$} is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:30: Lost braces; missing escapes or markup?
30 | \emph{$gamma=(gamma_1,gamma_2)$} is estimated by \emph{$tilde gamma=(tilde gamma_1,tilde gamma_2)$}, where \emph{$tilde gamma_1=1/n sum_{i=1}^n Z_i$} and
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:31: Lost braces; missing escapes or markup?
31 | \emph{$tilde gamma_2 = 1/(n-1) sum_{i=1}^{n-1}(Z_{n:i+1}-Z_{n:i})(H_{i+1}-H_i)$},
| ^
checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:32: Lost braces; missing escapes or markup?
32 | while \emph{$Z_{n:1}<= ... <= Z_{n:n}$} are ordered values of \emph{$Z_1, ..., Z_n$} and \emph{$H_i= phi^{-1}((i-3/8)(n+1/4))$}, cf. Chen and Shapiro (1995).
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.norm.test.Rd:35: Lost braces; missing escapes or markup?
35 | The pertaining matrix \emph{$[I^*(tilde gamma)]^{-1}$} does not depend on \emph{$tilde gamma$} and is calculated for succeding dimensions \emph{k} using some recurrent relations for Legendre's polynomials and is computed in a numerical way in case of cosine basis. In the implementation of \emph{$T^*$} the default value of \emph{c} is set to be 100. Therefore, in practice, \emph{$T^*$} is Schwarz-type criterion. See Inglot and Ledwina (2006) as well as Janic and Ledwina (2008) for comments. The resulting data driven test statistic for normality is \emph{$W_{T^*}=W_{T^*}(tilde gamma)$}.
| ^
checkRd: (-1) ddst.uniform.test.Rd:25: Lost braces; missing escapes or markup?
25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
| ^
checkRd: (-1) ddst.uniform.test.Rd:25: Lost braces; missing escapes or markup?
25 | $W_k=[1/sqrt(n) sum_{j=1}^k sum_{i=1}^n phi_j(Z_i)]^2$},
| ^
Check History
NOTE 0 OK · 14 NOTE · 0 WARNING · 0 ERROR · 0 FAILURE Mar 9, 2026
NOTE
r-devel-linux-x86_64-debian-clang
CRAN incoming feasibility
Maintainer: ‘Przemyslaw Biecek <przemyslaw.biecek@gmail.com>’
No Authors@R field in DESCRIPTION.
Please add one, modifying
Authors@R: c(person(given = "Przemyslaw",
family = "Biecek",
role = c("aut", "cre"),
email = "przemyslaw.biecek@gmail.com",
comment = "R code"),
person(given = "Teresa",
family = "Ledwina",
role = "aut",
c
NOTE
r-devel-linux-x86_64-debian-gcc
CRAN incoming feasibility
Maintainer: ‘Przemyslaw Biecek <przemyslaw.biecek@gmail.com>’
No Authors@R field in DESCRIPTION.
Please add one, modifying
Authors@R: c(person(given = "Przemyslaw",
family = "Biecek",
role = c("aut", "cre"),
email = "przemyslaw.biecek@gmail.com",
comment = "R code"),
person(given = "Teresa",
family = "Ledwina",
role = "aut",
c
NOTE
r-devel-linux-x86_64-fedora-clang
Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
2
NOTE
r-devel-linux-x86_64-fedora-gcc
Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
2
NOTE
r-devel-macos-arm64
Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
2
NOTE
r-devel-windows-x86_64
Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
2
NOTE
r-patched-linux-x86_64
Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
2
NOTE
r-release-linux-x86_64
Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
2
NOTE
r-release-macos-arm64
Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
2
NOTE
r-release-macos-x86_64
Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
2
NOTE
r-release-windows-x86_64
Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
2
NOTE
r-oldrel-macos-arm64
Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
2
NOTE
r-oldrel-macos-x86_64
Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
2
NOTE
r-oldrel-windows-x86_64
Rd files
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
29 | \emph{$W_k=[1/sqrt(n) sum_{i=1}^n l(Z_i)]I^{-1}[1/sqrt(n) sum_{i=1}^n l(Z_i)]'$},
| ^
checkRd: (-1) ddst-package.Rd:29: Lost braces; missing escapes or markup?
2
Reverse Dependencies (2)
depends
imports
Dependency Network
Version History
new
1.4
Mar 9, 2026