This function implements the two-sample PE simultaneous test on high-dimensional
mean vectors and covariance matrices using Cauchy 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 \(M_{PE}\) and \(T_{PE}\) denote
the PE mean test statistic and PE covariance test statistic, respectively.
(see meantest.pe.comp
and covtest.pe.comp for details).
Let \(p_{m}\) and \(p_{c}\) denote their respective \(p\)-values.
The PE simultaneous test statistic via Cauchy combination is defined as
$$C_{PE} = \frac{1}{2}\tan((0.5-p_{m})\pi) + \frac{1}{2}\tan((0.5-p_{c})\pi).$$
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 \(C_{PE}\) asymptotically converges in distribution to
a standard Cauchy distribution.
The asymptotic \(p\)-value is obtained by
$$p\text{-value} = 1-F_{Cauchy}(C_{PE}),$$
where \(F_{Cauchy}(\cdot)\) is the cdf of the standard Cauchy distribution.
simultest.pe.cauchy(dataX,dataY,delta_mean=NULL,delta_cov=NULL)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
a scalar; the thresholding value used in the construction of
the PE component for mean test; see meantest.pe.comp for details.
a scalar; the thresholding value used in the construction of
the PE component for covariance test; see covtest.pe.comp for details.
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.pe.cauchy(X,Y)
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