covtest.lc: Two-sample high-dimensional covariance test (Li and Chen, 2012)

Description

This function implements the two-sample $l_2$-norm-based high-dimensional covariance test proposed by Li and Chen (2012). 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 $\mathbf{Y}$. The test statistic $T_{LC}$ is defined as $$T_{LC} = A_{n_1}+B_{n_2}-2C_{n_1,n_2},$$ where $A_{n_1}$, $B_{n_2}$, and $C_{n_1,n_2}$ are unbiased estimators for $\mathrm{tr}(\mathbf{\Sigma}^2_1)$, $\mathrm{tr}(\mathbf{\Sigma}^2_2)$, and $\mathrm{tr}(\mathbf{\Sigma}_1\mathbf{\Sigma}_2)$, respectively. Under the null hypothesis $H_{0c}: \mathbf{\Sigma}_1 = \mathbf{\Sigma}_2$, the leading variance of $T_{LC}$ is $\sigma^2_{T_{LC}} = 4(\frac{1}{n_1}+\frac{1}{n_2})^2 \rm{tr}^2(\mathbf{\Sigma}^2)$, which can be consistently estimated by $\hat\sigma^2_{LC}$. The explicit formulas of $A_{n_1}$, $B_{n_2}$, $C_{n_1,n_2}$ and $\hat\sigma^2_{T_{LC}}$ can be found in Equations (2.1), (2.2) and Theorem 1 of Li and Chen (2012). With some regularity conditions, under the null hypothesis $H_{0c}: \mathbf{\Sigma}_1 = \mathbf{\Sigma}_2$, the test statistic $T_{LC}$ converges in distribution to a standard normal distribution as $n_1, n_2, p \rightarrow \infty$. The asymptotic $p$-value is obtained by $$p_{LC} = 1-\Phi(T_{LC}/\hat\sigma_{T_{LC}}),$$ where $\Phi(\cdot)$ is the cdf of the standard normal distribution.

Usage

covtest.lc(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

Li, J. and Chen, S. X. (2012). Two sample tests for high-dimensional covariance matrices. The Annals of Statistics, 40(2):908–940.

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.lc(X,Y)

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