| simultest.chisq {PEtests} | R Documentation |
Two-sample simultaneous test using chi-squared approximation
Description
This function implements the two-sample simultaneous test on high-dimensional
mean vectors and covariance matrices using chi-squared approximation.
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_{CQ}/\hat\sigma_{M_{CQ}} denote
the l_2-norm-based mean test statistic proposed in Chen and Qin (2010)
(see meantest.cq for details),
and let T_{LC}/\hat\sigma_{T_{LC}}
denote the l_2-norm-based covariance test statistic
proposed in Li and Chen (2012) (see covtest.lc for details).
The simultaneous test statistic via chi-squared approximation is defined as
S_{n_1, n_2} = M_{CQ}^2/\hat\sigma^2_{M_{CQ}} + T_{LC}^2/\hat\sigma^2_{T_{LC}}.
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 S_{n_1,n_2} asymptotically converges in distribution to
a \chi_2^2 distribution.
The asymptotic p-value is obtained by
p\text{-value} = 1-F_{\chi_2^2}(S_{n_1,n_2}),
where F_{\chi_2^2}(\cdot) is the cdf of the \chi_2^2 distribution.
Usage
simultest.chisq(dataX,dataY)
Arguments
dataX |
n1 by p data matrix |
dataY |
n2 by p data matrix |
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.chisq(X,Y)