This function implements the two-sample PE covariance test via
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 \(p_{CQ}\) and \(p_{CLX}\) denote the \(p\)-values associated with
the \(l_2\)-norm-based covariance test (see meantest.cq for details)
and the \(l_\infty\)-norm-based covariance test
(see meantest.clx for details), respectively.
The PE covariance test via Cauchy combination is defined as
$$M_{Cauchy} = \frac{1}{2}\tan((0.5-p_{CQ})\pi) + \frac{1}{2}\tan((0.5-p_{CLX})\pi).$$
It has been proved that with some regularity conditions, under the null hypothesis
\(H_{0m}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2,\)
the two tests are asymptotically independent as \(n_1, n_2, p\rightarrow \infty\),
and therefore \(M_{Cauchy}\) asymptotically converges in distribution to a standard Cauchy distribution.
The asymptotic \(p\)-value is obtained by
$$p\text{-value} = 1-F_{Cauchy}(M_{Cauchy}),$$
where \(F_{Cauchy}(\cdot)\) is the cdf of the standard Cauchy distribution.