covtest.pe.cauchy {PEtests}

Two-sample PE covariance test for high-dimensional data via Cauchy combination

Description

This function implements the two-sample PE covariance test via Cauchy combination. 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}. Let p_{LC} and p_{CLX} denote the p-values associated with the l_2-norm-based covariance test (see covtest.lc for details) and the l_\infty-norm-based covariance test (see covtest.clx for details), respectively. The PE covariance test via Cauchy combination is defined as

T_{Cauchy} = \frac{1}{2}\tan((0.5-p_{LC})\pi) + \frac{1}{2}\tan((0.5-p_{CLX})\pi).

It has been proved that with some regularity conditions, under the null hypothesis H_{0c}: \mathbf{\Sigma}_1 = \mathbf{\Sigma}_2, the two tests are asymptotically independent as n_1, n_2, p\rightarrow \infty, and therefore T_{Cauchy} asymptotically converges in distribution to a standard Cauchy distribution. The asymptotic p-value is obtained by

p\text{-value} = 1-F_{Cauchy}(T_{Cauchy}),

where F_{Cauchy}(\cdot) is the cdf of the standard Cauchy distribution.

Usage

covtest.pe.cauchy(dataX,dataY)

Arguments

dataX

an n_1 by p data matrix

dataY

an n_2 by p data matrix

Value

stat the value of test statistic

pval the p-value for the test.

References

Yu, X., Li, D., and Xue, L. (2022). Fisher’s combined probability test for high-dimensional covariance matrices. 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.cauchy(X,Y)