Suppose we have two independent high-dimensional samples:
$$
\boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma}_i,\; i=1,2.
$$
The primary object is to test
$$H_{0}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2\; \operatorname{versus}\; H_{1}: \boldsymbol{\mu}_1 \neq \boldsymbol{\mu}_2.$$
Zhu et al. (2023) proposed the following test statistic:
$$T_{ZWZ}=\frac{n_1n_2n^{-1}\|\bar{\boldsymbol{y}}_1-\bar{\boldsymbol{y}}_2\|^2}{\operatorname{tr}(\hat{\boldsymbol{\Omega}}_n)},$$
where \(\bar{\boldsymbol{y}}_{i},i=1,2\) are the sample mean vectors and \(\hat{\boldsymbol{\Omega}}_n\) is the estimator of \(\operatorname{Cov}[(n_1n_2/n)^{1/2}(\bar{\boldsymbol{y}}_1-\bar{\boldsymbol{y}}_2)]\).
They showed that under the null hypothesis, \(T_{ZWZ}\) and an F-type mixture have the same normal or non-normal limiting distribution.