simultest.pe.fisher {PEtests}

Two-sample PE simultaneous test using Fisher's combination

Description

This function implements the two-sample PE simultaneous test on high-dimensional mean vectors and covariance matrices using Fisher's 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 Fisher's combination is defined as

J_{PE} = -2\log(p_{m})-2\log(p_{c}).

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 J_{PE} asymptotically converges in distribution to a \chi_4^2 distribution. The asymptotic p-value is obtained by

p\text{-value} = 1-F_{\chi_4^2}(J_{PE}),

where F_{\chi_4^2}(\cdot) is the cdf of the \chi_4^2 distribution.

Usage

simultest.pe.fisher(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.fisher(X,Y)