simultest.pe.cauchy {PEtests}

Two-sample PE simultaneous test using Cauchy combination

Description

This function implements the two-sample PE simultaneous test on high-dimensional mean vectors and covariance matrices using 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 M_{PE} and T_{PE} denote the PE mean test statistic and PE covariance test statistic, respectively. (see meantest.pe.comp and covtest.pe.comp for details). Let p_{m} and p_{c} denote their respective p-values. The PE simultaneous test statistic via Cauchy combination is defined as

C_{PE} = \frac{1}{2}\tan((0.5-p_{m})\pi) + \frac{1}{2}\tan((0.5-p_{c})\pi).

It has been proved that with some regularity conditions, under the null hypothesis H_0: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2 \ \text{ and } \ \mathbf{\Sigma}_1 = \mathbf{\Sigma}_2, the two tests are asymptotically independent as n_1, n_2, p\rightarrow \infty, and therefore C_{PE} asymptotically converges in distribution to a standard Cauchy distribution. The asymptotic p-value is obtained by

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

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

Usage

simultest.pe.cauchy(dataX,dataY,delta_mean=NULL,delta_cov=NULL)

Arguments

dataX

an n_1 by p data matrix

dataY

an n_2 by p data matrix

delta_mean

a scalar; the thresholding value used in the construction of the PE component for mean test; see meantest.pe.comp for details.

delta_cov

a scalar; the thresholding value used in the construction of the PE component for covariance test; see covtest.pe.comp for details.

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)
simultest.pe.cauchy(X,Y)