NRAHDLTP (version 0.1.2)

tsbf_skk2013: Test proposed by Srivastava et al. (2013)

Description

Srivastava et al. (2013)'s test for testing equality of two-sample high-dimensional mean vectors without assuming that two covariance matrices are the same.

Usage

tsbf_skk2013(y1, y2)

Value

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

statistic

the test statistic proposed by Srivastava et al. (2013)

p.value

the \(p\)-value of the test proposed by Srivastava et al. (2013)

cpn

the adjustment coefficient proposed by Srivastava et al. (2013)

Arguments

y1

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

y2

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

Details

Suppose we have two independent high-dimensional samples: $$ \boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma}_i,i=1,2. $$ The primary object is to test $$H_{0}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2\; \operatorname{versus}\; H_{1}: \boldsymbol{\mu}_1 \neq \boldsymbol{\mu}_2.$$ Srivastava et al. (2013) proposed the following test statistic: $$T_{SKK} = \frac{(\bar{\boldsymbol{y}}_1 - \bar{\boldsymbol{y}}_2)^\top \hat{\boldsymbol{D}}^{-1}(\bar{\boldsymbol{y}}_1 - \bar{\boldsymbol{y}}_2) - p}{\sqrt{2 \widehat{\operatorname{Var}}(\hat{q}_n) c_{p,n}}},$$ where \(\bar{\boldsymbol{y}}_{i},i=1,2\) are the sample mean vectors, \(\hat{\boldsymbol{D}}=\hat{\boldsymbol{D}}_1/n_1+\hat{\boldsymbol{D}}_2/n_2\) with \(\hat{\boldsymbol{D}}_i,i=1,2\) being the diagonal matrices consisting of only the diagonal elements of the sample covariance matrices. \(\widehat{\operatorname{Var}}(\hat{q}_n)\) is given by equation (1.18) in Srivastava et al. (2013), and \(c_{p, n}\) is the adjustment coefficient proposed by Srivastava et al. (2013). They showed that under the null hypothesis, \(T_{SKK}\) is asymptotically normally distributed.

References

Srivastava_2013NRAHDLTP

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_skk2013(y1, y2)