meantest.pe.comp: Two-sample PE mean test for high-dimensional data via PE component
Description
This function implements the two-sample PE mean via the
construction of the PE component. Let \(M_{CQ}/\hat\sigma_{M_{CQ}}\)
denote the \(l_2\)-norm-based mean test statistic
(see meantest.cq for details).
The PE component is constructed by
$$J_m = \sqrt{p}\sum_{i=1}^p M_i\widehat\nu^{-1/2}_i
\mathcal{I}\{ \sqrt{2}M_i\widehat\nu^{-1/2}_i + 1 > \delta_{mean} \}, $$
where \(\delta_{mean}\) is a threshold for the screening procedure,
recommended to take the value of \(\delta_{mean}=2\log(\log (n_1+n_2))\log p\).
The explicit forms of \(M_{i}\) and \(\widehat\nu_{j}\)
can be found in Section 3.1 of Yu et al. (2022).
The PE covariance test statistic is defined as
$$M_{PE}=M_{CQ}/\hat\sigma_{M_{CQ}}+J_m.$$
With some regularity conditions, under the null hypothesis
\(H_{0m}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2\),
the test statistic \(M_{PE}\) converges in distribution to
a standard normal distribution as \(n_1, n_2, p \rightarrow \infty\).
The asymptotic \(p\)-value is obtained by
$$p\text{-value}= 1-\Phi(M_{PE}),$$
where \(\Phi(\cdot)\) is the cdf of the standard normal distribution.
Usage
meantest.pe.comp(dataX,dataY,delta=NULL)
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
delta
a scalar; the thresholding value used in the construction of
the PE component. If not specified, the function uses a default value
\(\delta_{mean}=2\log(\log (n_1+n_2))\log p\).
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.