meantest.pe.comp {PEtests}

Two-sample PE mean test for high-dimensional data via PE component

Description

This function implements the two-sample PE mean via the construction of the PE component. Let M_{CQ}/\hat\sigma_{M_{CQ}} denote the l_2-norm-based mean test statistic (see meantest.cq for details). The PE component is constructed by

J_m = \sqrt{p}\sum_{i=1}^p M_i\widehat\nu^{-1/2}_i \mathcal{I}\{ \sqrt{2}M_i\widehat\nu^{-1/2}_i + 1 > \delta_{mean} \},

where \delta_{mean} is a threshold for the screening procedure, recommended to take the value of \delta_{mean}=2\log(\log (n_1+n_2))\log p. The explicit forms of M_{i} and \widehat\nu_{j} can be found in Section 3.1 of Yu et al. (2022). The PE covariance test statistic is defined as

M_{PE}=M_{CQ}/\hat\sigma_{M_{CQ}}+J_m.

With some regularity conditions, under the null hypothesis H_{0m}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2, the test statistic M_{PE} converges in distribution to a standard normal distribution as n_1, n_2, p \rightarrow \infty. The asymptotic p-value is obtained by

p\text{-value}= 1-\Phi(M_{PE}),

where \Phi(\cdot) is the cdf of the standard normal distribution.

Usage

meantest.pe.comp(dataX,dataY,delta=NULL)

Arguments

dataX

an n_1 by p data matrix

dataY

an n_2 by p data matrix

delta

a scalar; the thresholding value used in the construction of the PE component. If not specified, the function uses a default value \delta_{mean}=2\log(\log (n_1+n_2))\log p.

Value

stat the value of test statistic

pval the p-value for the test.

References

Yu, X., Li, D., Xue, L., and Li, R. (2022). Power-enhanced simultaneous test of high-dimensional mean vectors and covariance matrices with application to gene-set testing. Journal of the American Statistical Association, (in press):1–14.

Examples

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.pe.comp(X,Y)