covtest.pe.comp {PEtests}

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

Description

This function implements the two-sample PE covariance test via the construction of the PE component. Let T_{LC}/\hat\sigma_{T_{LC}} denote the l_2-norm-based covariance test statistic (see covtest.lc for details). The PE component is constructed by

J_c=\sqrt{p}\sum_{i=1}^p\sum_{j=1}^p T_{ij}\widehat\xi^{-1/2}_{ij} \mathcal{I}\{ \sqrt{2}T_{ij}\widehat\xi^{-1/2}_{ij} +1 > \delta_{cov} \},

where \delta_{cov} is a threshold for the screening procedure, recommended to take the value of \delta_{cov}=4\log(\log (n_1+n_2))\log p. The explicit forms of T_{ij} and \widehat\xi_{ij} can be found in Section 3.2 of Yu et al. (2022). The PE covariance test statistic is defined as

T_{PE}=T_{LC}/\hat\sigma_{T_{LC}}+J_c.

With some regularity conditions, under the null hypothesis H_{0c}: \mathbf{\Sigma}_1 = \mathbf{\Sigma}_2, the test statistic T_{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(T_{PE}),

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

Usage

covtest.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_{cov}=4\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)
covtest.pe.comp(X,Y)