NRAHDLTP (version 0.1.2)

ts_zzz2020: Test proposed by Zhang et al. (2020)

Description

Zhang et al. (2020)'s test for testing equality of two-sample high-dimensional mean vectors with assuming that two covariance matrices are the same.

Usage

ts_zzz2020(y1, y2)

Value

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

p.value

the p-value of the test proposed by Zhang et al. (2020).

statistic

the test statistic proposed by Zhang et al. (2020).

df

estimated approximate degrees of freedom of Zhang et al. (2020)'s test.

Arguments

y1

The data matrix (\(p\) by \(n_1\)) from the first population. Each column represents a \(p\)-dimensional sample.

y2

The data matrix (\(p\) by \(n_2\)) from the first population. Each column represents a \(p\)-dimensional sample.

Details

Suppose we have two independent high-dimensional samples: $$ \boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma},i=1,2. $$ The primary object is to test $$H_{0}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2\; \operatorname{versus}\; H_{1}: \boldsymbol{\mu}_1 \neq \boldsymbol{\mu}_2.$$ Zhang et al.(2020) proposed the following test statistic: $$T_{ZZZ} = \frac{n_1n_2}{(n_1+n_2)p}(\bar{\boldsymbol{y}}_1 - \bar{\boldsymbol{y}}_2)^\top \hat{\boldsymbol{D}}^{-1}(\bar{\boldsymbol{y}}_1 - \bar{\boldsymbol{y}}_2),$$ where \(\bar{\boldsymbol{y}}_{i},i=1,2\) are the sample mean vectors, \(\hat{\boldsymbol{D}}\) is the diagonal matrix of sample covariance matrix. They showed that under the null hypothesis, \(T_{ZZZ}\) and a chi-squared-type mixture have the same limiting distribution.

References

Zhang_2020NRAHDLTP

Examples

set.seed(1234)
n1 <- 20
n2 <- 30
p <- 50
mu1 <- t(t(rep(0, p)))
mu2 <- mu1
rho <- 0.1
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)
Z1 <- matrix(rnorm(n1 * p, mean = 0, sd = 1), p, n1)
Z2 <- matrix(rnorm(n2 * p, mean = 0, sd = 1), p, n2)
y1 <- Gamma %*% Z1 + mu1 %*% (rep(1, n1))
y2 <- Gamma %*% Z2 + mu2 %*% (rep(1, n2))
ts_zzz2020(y1, y2)