covtest.pe.fisher: Two-sample PE covariance test for high-dimensional data via Fisher's combination

Description

This function implements the two-sample PE covariance test via 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 $p_{LC}$ 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 Fisher's combination is defined as $$T_{Fisher} = -2\log(p_{LC})-2\log(p_{CLX}).$$ 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 $T_{Fisher}$ asymptotically converges in distribution to a $\chi_4^2$ distribution. The asymptotic $p$-value is obtained by $$p\text{-value} = 1-F_{\chi_4^2}(T_{Fisher}),$$ where $F_{\chi_4^2}(\cdot)$ is the cdf of the $\chi_4^2$ distribution.

Usage

covtest.pe.fisher(dataX,dataY)

Value

stat the value of test statistic

pval the p-value for the test.

Arguments

dataX: an $n_1$ by $p$ data matrix
dataY: an $n_2$ by $p$ data matrix

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

Run this code

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.fisher(X,Y)

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