glhtbf_zz2022 {HDNRA}R Documentation

Test 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

glhtbf_zz2022(Y,G,n,p)

Arguments

Y

A list of kk data matrices. The iith element represents the data matrix (p×nip\times n_i) from the iith population with each column representing a pp-dimensional observation.

G

A known full-rank coefficient matrix (q×kq\times k) with rank(G)<k\operatorname{rank}(\boldsymbol{G})< k.

n

A vector of kk sample sizes. The iith element represents the sample size of group ii, nin_i.

p

The dimension of data.

Details

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

yi1,,yini,  are  i.i.d.  with  E(yi1)=μi,  Cov(yi1)=Σi,i=1,,k. \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:

H0:GM=0, vs. H1:GM0,H_0: \boldsymbol{G M}=\boldsymbol{0}, \quad \text { vs. } H_1: \boldsymbol{G M} \neq \boldsymbol{0},

where M=(μ1,,μk)\boldsymbol{M}=(\boldsymbol{\mu}_1,\ldots,\boldsymbol{\mu}_k)^\top is a k×pk\times p matrix collecting kk mean vectors and G:q×k\boldsymbol{G}:q\times k is a known full-rank coefficient matrix with rank(G)<k\operatorname{rank}(\boldsymbol{G})<k.

Let yˉi,i=1,,k\bar{\boldsymbol{y}}_{i},i=1,\ldots,k be the sample mean vectors and Σ^i,i=1,,k\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:

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

where C=[(GDG)1/2G]Ip\boldsymbol{C}=[(\boldsymbol{G D G}^\top)^{-1/2}\boldsymbol{G}]\otimes\boldsymbol{I}_p, D=diag(1/n1,,1/nk)\boldsymbol{D}=\operatorname{diag}(1/n_1,\ldots,1/n_k), and hijh_{ij} is the (i,j)(i,j)th entry of the k×kk\times k matrix H=G(GDG)1G\boldsymbol{H}=\boldsymbol{G}^\top(\boldsymbol{G}\boldsymbol{D}\boldsymbol{G}^\top)^{-1}\boldsymbol{G}.

Value

A (list) object of S3 class htest containing the following elements:

p.value

the pp-value of the test proposed by Zhang and Zhu (2022).

statistic

the test statistic proposed by Zhang and Zhu (2022).

beta0

the parameter used in Zhang and Zhu (2022)'s test.

beta1

the parameter used in Zhang and Zhu (2022)'s test.

df

estimated approximate degrees of freedom of Zhang and Zhu (2022)'s test.

References

Zhang J, Zhu T (2022). “A new normal reference test for linear hypothesis testing in high-dimensional heteroscedastic one-way MANOVA.” Computational Statistics & Data Analysis, 168, 107385. doi:10.1016/j.csda.2021.107385.

Examples

set.seed(1234)
k <- 3
p <- 50
n <- c(25, 30, 40)
rho <- 0.1
M <- matrix(rep(0, k * p), nrow = k, ncol = p)
avec <- seq(1, k)
Y <- list()
for (g in 1:k) {
  a <- avec[g]
  y <- (-2 * sqrt(a * (1 - rho)) + sqrt(4 * a * (1 - rho) + 4 * p * a * rho)) / (2 * p)
  x <- y + sqrt(a * (1 - rho))
  Gamma <- matrix(rep(y, p * p), nrow = p)
  diag(Gamma) <- rep(x, p)
  Z <- matrix(rnorm(n[g] * p, mean = 0, sd = 1), p, n[g])
  Y[[g]] <- Gamma %*% Z + t(t(M[g, ])) %*% (rep(1, n[g]))
}
G <- cbind(diag(k - 1), rep(-1, k - 1))
glhtbf_zz2022(Y, G, n, p)


[Package HDNRA version 1.0.0 Index]