glhtbf_zz2022 {HDNRA}

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 k data matrices. The ith element represents the data matrix (p\times n_i) from the ith population with each column 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) object of S3 class htest containing the following elements:

p.value

the p-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)