ZZ2022.GLHTBF.3cNRT: Normal-reference-test with three-cumulant (3-c) matched...

In HDNRA: High-Dimensional Location Testing with Normal-Reference Approaches

Normal-reference-test with three-cumulant (3-c) matched $\chi^2$-approximation for GLHT problem under heteroscedasticity proposed by Zhang and Zhu (2022)

Description

Zhang and Zhu (2022)'s test for general linear hypothesis testing (GLHT) problem for high-dimensional data under heteroscedasticity.

Usage

ZZ2022.GLHTBF.3cNRT(Y,G,n,p)

Arguments

Y	A list of k data matrices. The ith element represents the data matrix (n_i\times p) from the ith population with each row representing a p-dimensional observation.
G	A known full-rank coefficient matrix (q\times k) with \operatorname{rank}(\boldsymbol{G})< k.
n	A vector of k sample sizes. The ith element represents the sample size of group i, n_i.
p	The dimension of data.

Details

Suppose we have the following k independent high-dimensional samples:

\boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma}_i,i=1,\ldots,k.

It is of interest to test the following GLHT problem:

H_0: \boldsymbol{G M}=\boldsymbol{0}, \quad \text { vs. } H_1: \boldsymbol{G M} \neq \boldsymbol{0},

where \boldsymbol{M}=(\boldsymbol{\mu}_1,\ldots,\boldsymbol{\mu}_k)^\top is a k\times p matrix collecting k mean vectors and \boldsymbol{G}:q\times k is a known full-rank coefficient matrix with \operatorname{rank}(\boldsymbol{G})<k.

Let \bar{\boldsymbol{y}}_{i},i=1,\ldots,k be the sample mean vectors and \hat{\boldsymbol{\Sigma}}_i,i=1,\ldots,k be the sample covariance matrices.

Zhang and Zhu (2022) proposed the following U-statistic based test statistic:

T_{ZZ}=\|\boldsymbol{C \hat{\mu}}\|^2-\sum_{i=1}^kh_{ii}\operatorname{tr}(\hat{\boldsymbol{\Sigma}}_i)/n_i,

where \boldsymbol{C}=[(\boldsymbol{G D G}^\top)^{-1/2}\boldsymbol{G}]\otimes\boldsymbol{I}_p, \boldsymbol{D}=\operatorname{diag}(1/n_1,\ldots,1/n_k), and h_{ij} is the (i,j)th entry of the k\times k matrix \boldsymbol{H}=\boldsymbol{G}^\top(\boldsymbol{G}\boldsymbol{D}\boldsymbol{G}^\top)^{-1}\boldsymbol{G}.

Value

A list of class "NRtest" containing the results of the hypothesis test. See the help file for NRtest.object for details.

References

\insertRef

Zhang_2022heteroscedasticHDNRA

Examples

library("HDNRA")
data("corneal")
dim(corneal)
group1 <- as.matrix(corneal[1:43, ]) ## normal group
group2 <- as.matrix(corneal[44:57, ]) ## unilateral suspect group
group3 <- as.matrix(corneal[58:78, ]) ## suspect map group
group4 <- as.matrix(corneal[79:150, ]) ## clinical keratoconus group
p <- dim(corneal)[2]
Y <- list()
k <- 4
Y[[1]] <- group1
Y[[2]] <- group2
Y[[3]] <- group3
Y[[4]] <- group4
n <- c(nrow(Y[[1]]),nrow(Y[[2]]),nrow(Y[[3]]),nrow(Y[[4]]))
G <- cbind(diag(k-1),rep(-1,k-1))
ZZ2022.GLHTBF.3cNRT(Y,G,n,p)

HDNRA documentation built on Oct. 30, 2024, 9:28 a.m.