NRAHDLTP (version 0.1.2)

glht_zgz2017: Test proposed by Zhang et al. (2017)

Description

Zhang et al. (2017)'s test for general linear hypothesis testing (GLHT) problem for high-dimensional data with assuming that underlying covariance matrices are the same.

Usage

glht_zgz2017(Y,G,n,p)

Value

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

statistic

the test statistic proposed by Zhang et al. (2017)

p.value

the \(p\)-value of the test proposed by Zhang et al. (2017).

df

estimated approximate degrees of freedom of Zhang et al.(2017)'s test.

Arguments

Y

A list of \(k\) data matrices. The \(i\)th element represents the data matrix (\(p\times n_i\)) from the \(i\)th population with each column representing a \(p\)-dimensional sample.

G

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

n

A vector of \(k\) sample sizes. The \(i\)th 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: $$ \boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma},\;i=1,\ldots,k. $$ It is of interest to test the following GLHT problem: $$H_0: \boldsymbol{G M}=\boldsymbol{0}, \quad \text { vs. } \quad H_1: \boldsymbol{G M} \neq \boldsymbol{0},$$ 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 \(\operatorname{rank}(\boldsymbol{G})<k\).

Zhang et al. (2017) proposed the following test statistic: $$ T_{ZGZ}=\|\boldsymbol{C \hat{\mu}}\|^2, $$ where \(\boldsymbol{C}=[(\boldsymbol{G D G}^\top)^{-1/2}\boldsymbol{G}]\otimes\boldsymbol{I}_p\), and \(\hat{\boldsymbol{\mu}}=(\bar{\boldsymbol{y}}_1^\top,\ldots,\bar{\boldsymbol{y}}_k^\top)^\top\), with \(\bar{\boldsymbol{y}}_{i},i=1,\ldots,k\) being the sample mean vectors and \(\boldsymbol{D}=\operatorname{diag}(1/n_1,\ldots,1/n_k)\).

They showed that under the null hypothesis, \(T_{ZGZ}\) and a chi-squared-type mixture have the same normal or non-normal limiting distribution.

References

Zhang_2017NRAHDLTP

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)
y <- (-2 * sqrt(1 - rho) + sqrt(4 * (1 - rho) + 4 * p * rho)) / (2 * p)
x <- y + sqrt((1 - rho))
Gamma <- matrix(rep(y, p * p), nrow = p)
diag(Gamma) <- rep(x, p)
Y <- list()
for (g in 1:k) {
  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))
glht_zgz2017(Y, G, n, p)