# apval_Sri2008: Asymptotics-Based p-value of the Test Proposed by Srivastava... In highmean: Two-Sample Tests for High-Dimensional Mean Vectors

## Description

Calculates p-value of the test for testing equality of two-sample high-dimensional mean vectors proposed by Srivastava and Du (2008) based on the asymptotic distribution of the test statistic.

## Usage

 1 apval_Sri2008(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.

## 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}: μ_1 = μ_2 versus H_{A}: μ_1 \neq μ_2. Let \bar{X}_{k} be the sample mean for group k = 1, 2. Also, let S = n^{-1} ∑_{k = 1}^{2} ∑_{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.

Srivastava and Du (2008) proposed the following test statistic:

T_{SD} = \frac{(n_1^{-1} + n_2^{-1})^{-1} (\bar{X}_1 - \bar{X}_2)^T D_S^{-1} (\bar{X}_1 - \bar{X}_2) - (n - 2)^{-1} n p}{√{2 (tr R^2 - p^2 n^{-1}) c_{p, n}}},

where D_S = diag (s_{11}, s_{22}, ..., s_{pp}), s_{ii}'s are the diagonal elements of S, R = D_S^{-1/2} S D_S^{-1/2} is the sample correlation matrix and c_{p, n} = 1 + tr R^2 p^{-3/2}. This test statistic follows normal distribution under the null hypothesis.

## 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 Srivastava and Du (2008).

## Note

The asymptotic distribution of the test statistic was derived under normality assumption in Bai and Saranadasa (1996). Also, this function assumes that the two sample populations have a common covariance matrix.

## References

Srivastava MS and Du M (2008). "A test for the mean vector with fewer observations than the dimension." Journal of Multivariate Analysis, 99(3), 386–402.

## Examples

 1 2 3 4 5 6 7 8 9 10 11 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_Sri2008(sam1, sam2)

### Example output

Loading required package: mvtnorm
$sam.info n1 n2 p 50 50 200$cov.assumption
[1] "the two groups have same covariance"

$method [1] "asymptotic distribution"$pval
Sri2008
0.61321

highmean documentation built on May 2, 2019, 3:45 p.m.