tsbf_zwz2023 {HDNRA}

Test proposed by Zhu et al. (2023)

Description

Zhu et al. (2023)'s test for testing equality of two-sample high-dimensional mean vectors without assuming that two covariance matrices are the same.

Usage

tsbf_zwz2023(y1, y2)

Arguments

y1

The data matrix (p by n1) from the first population. Each column represents a p-dimensional observation.

y2

The data matrix (p by n2) from the first population. Each column represents a p-dimensional observation.

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,\; 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.

Zhu et al. (2023) proposed the following test statistic:

T_{ZWZ}=\frac{n_1n_2n^{-1}\|\bar{\boldsymbol{y}}_1-\bar{\boldsymbol{y}}_2\|^2}{\operatorname{tr}(\hat{\boldsymbol{\Omega}}_n)},

where \bar{\boldsymbol{y}}_{i},i=1,2 are the sample mean vectors and \hat{\boldsymbol{\Omega}}_n is the estimator of \operatorname{Cov}[(n_1n_2/n)^{1/2}(\bar{\boldsymbol{y}}_1-\bar{\boldsymbol{y}}_2)]. They showed that under the null hypothesis, T_{ZWZ} and an F-type mixture have the same normal or non-normal limiting distribution.

Value

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

p.value

the p-value of the test proposed by Zhu et al. (2023).

statistic

the test statistic proposed by Zhu et al. (2023).

df1

estimated approximate degrees of freedom d_1 of Zhu et al. (2023)'s test.

df2

estimated approximate degrees of freedom d_2 of Zhu et al. (2023)'s test.

References

Zhu T, Wang P, Zhang J (2023). “Two-sample Behrens–Fisher problems for high-dimensional data: a normal reference F-type test.” Computational Statistics, 1–24. doi:10.1007/s00180-023-01433-6.

Examples

set.seed(1234)
n1 <- 20
n2 <- 30
p <- 50
mu1 <- t(t(rep(0, p)))
mu2 <- mu1
rho1 <- 0.1
rho2 <- 0.2
a1 <- 1
a2 <- 2
w1 <- (-2 * sqrt(a1 * (1 - rho1)) + sqrt(4 * a1 * (1 - rho1) + 4 * p * a1 * rho1)) / (2 * p)
x1 <- w1 + sqrt(a1 * (1 - rho1))
Gamma1 <- matrix(rep(w1, p * p), nrow = p)
diag(Gamma1) <- rep(x1, p)
w2 <- (-2 * sqrt(a2 * (1 - rho2)) + sqrt(4 * a2 * (1 - rho2) + 4 * p * a2 * rho2)) / (2 * p)
x2 <- w2 + sqrt(a2 * (1 - rho2))
Gamma2 <- matrix(rep(w2, p * p), nrow = p)
diag(Gamma2) <- rep(x2, p)
Z1 <- matrix(rnorm(n1*p,mean = 0,sd = 1), p, n1)
Z2 <- matrix(rnorm(n2*p,mean = 0,sd = 1), p, n2)
y1 <- Gamma1 %*% Z1 + mu1%*%(rep(1,n1))
y2 <- Gamma2 %*% Z2 + mu2%*%(rep(1,n2))
tsbf_zwz2023(y1, y2)