highmean (version 2.0)

apval_Bai1996: Asymptotics-Based p-value of the Test Proposed by Bai and Saranadasa (1996)

Description

Calculates p-value of the test for testing equality of two-sample high-dimensional mean vectors proposed by Bai and Saranadasa (1996) based on the asymptotic distribution of the test statistic.

Usage

apval_Bai1996(sam1, sam2)

Arguments

sam1

an n1 by p matrix from sample population 1. Each row represents a $p$-dimensional sample.

sam2

an n2 by p matrix from sample population 2. Each row represents a $p$-dimensional sample.

Value

• A list including the following elements:
• sam.info
  the basic information about the two groups of samples, including the samples sizes and dimension.
• cov.assumption
  this output reminds users that the two sample populations have a common covariance matrix.
• method
  this output reminds users that the p-values are obtained using the asymptotic distributions of test statistics.
• pval
  the p-value of the test proposed by Bai and Saranadasa (1996).

Details

Suppose that the two groups of $p$-dimensional independent and identically distributed samples ${X_{1i}}_{i=1}^{n_1}$ and ${X_{2j}}_{j=1}^{n_2}$ are observed; we consider high-dimensional data with $p \gg n := n_1 + n_2 - 2$. Assume that the two groups share a common covariance matrix. The primary object is to test $H_{0}: \mu_1 = \mu_2$ versus $H_{A}: \mu_1 \neq \mu_2$. Let $\bar{X}_{k}$ be the sample mean for group $k = 1, 2$. Also, let $S = n^{-1} \sum_{k = 1}^{2} \sum_{i = 1}^{n_{k}} (X_{ki} - \bar{X}_k) (X_{ki} - \bar{X}_k)^T$ be the pooled sample covariance matrix from the two groups. Bai and Saranadasa (1996) proposed the following test statistic: $$T_{BS} = \frac{(n_1^{-1} + n_2^{-1})^{-1} (\bar{X}_1 - \bar{X}_2)^T (\bar{X}_1 - \bar{X}_2) - tr S}{\sqrt{2 n (n + 1) (n - 1)^{-1} (n + 2)^{-1} [tr S^2 - n^{-1} (tr S)^2]}},$$ and its asymptotic distribution is normal under the null hypothesis.

References

Bai ZD and Saranadasa H (1996). "Effect of high dimension: by an example of a two sample problem." Statistica Sinica, 6(2), 311--329.

Examples

library(MASS)
set.seed(1234)
n1 <- n2 <- 50
p <- 200
mu1 <- rep(0, p)
mu2 <- mu1
mu2[1:10] <- 0.2
true.cov <- 0.4^(abs(outer(1:p, 1:p, "-"))) # AR1 covariance
sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov)
sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov)
apval_Bai1996(sam1, sam2)