meantest.clx: Two-sample high-dimensional mean test (Cai, Liu and Xia, 2014)

Description

This function implements the two-sample $l_\infty$-norm-based high-dimensional mean test proposed in Cai, Liu and Xia (2014). Suppose $\{\mathbf{X}_1, \ldots, \mathbf{X}_{n_1}\}$ are i.i.d. copies of $\mathbf{X}$, and $\{\mathbf{Y}_1, \ldots, \mathbf{Y}_{n_2}\}$ are i.i.d. copies of $\mathbf{Y}$. The test statistic is defined as $$ M_{CLX}=\frac{n_1n_2}{n_1+n_2}\max_{1\leq j\leq p} \frac{(\bar{X_j}-\bar{Y_j})^2} {\frac{1}{n_1+n_2} [\sum_{u=1}^{n_1} (X_{uj}-\bar{X_j})^2+\sum_{v=1}^{n_2} (Y_{vj}-\bar{Y_j})^2] } $$ With some regularity conditions, under the null hypothesis $H_{0c}: \mathbf{\Sigma}_1 = \mathbf{\Sigma}_2$, the test statistic $M_{CLX}-2\log p+\log\log p$ converges in distribution to a Gumbel distribution $G_{mean}(x) = \exp(-\frac{1}{\sqrt{\pi}}\exp(-\frac{x}{2}))$ as $n_1, n_2, p \rightarrow \infty$. The asymptotic $p$-value is obtained by $$p_{CLX} = 1-G_{mean}(M_{CLX}-2\log p+\log\log p).$$

Usage

meantest.clx(dataX,dataY)

Value

stat the value of test statistic

pval the p-value for the test.

Arguments

dataX: an $n_1$ by $p$ data matrix
dataY: an $n_2$ by $p$ data matrix

References

Cai, T. T., Liu, W., and Xia, Y. (2014). Two-sample test of high dimensional means under dependence. Journal of the Royal Statistical Society: Series B: Statistical Methodology, 76(2):349–372.

Examples

Run this code

n1 = 100; n2 = 100; pp = 500
set.seed(1)
X = matrix(rnorm(n1*pp), nrow=n1, ncol=pp)
Y = matrix(rnorm(n2*pp), nrow=n2, ncol=pp)
meantest.clx(X,Y)