This function implements the two-sample 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 \(p_{CQ}\) and \(p_{LC}\) denote the \(p\)-values associated with
the \(l_2\)-norm-based mean test proposed in Chen and Qin (2010)
(see meantest.cq for details)
and the \(l_2\)-norm-based covariance test proposed in Li and Chen (2012)
(see covtest.lc for details),
respectively.
The simultaneous test statistic via Fisher's combination is defined as
$$J_{n_1, n_2} = -2\log(p_{CQ}) -2\log(p_{LC}).$$
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 \(J_{n_1,n_2}\) asymptotically converges in distribution to
a \(\chi_4^2\) distribution.
The asymptotic \(p\)-value is obtained by
$$p\text{-value} = 1-F_{\chi_4^2}(J_{n_1,n_2}),$$
where \(F_{\chi_4^2}(\cdot)\) is the cdf of the \(\chi_4^2\) distribution.
simultest.fisher(dataX,dataY)stat the value of test statistic
pval the p-value for the test.
an \(n_1\) by \(p\) data matrix
an \(n_2\) by \(p\) data matrix
Chen, S. X. and Qin, Y. L. (2010). A two-sample test for high-dimensional data with applications to gene-set testing. Annals of Statistics, 38(2):808–835.
Li, J. and Chen, S. X. (2012). Two sample tests for high-dimensional covariance matrices. The Annals of Statistics, 40(2):908–940.
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.
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.fisher(X,Y)
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