Suppose we have two independent high-dimensional samples:
$$
\boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma},i=1,2.
$$
The primary object is to test
$$H_{0}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2\; \operatorname{versus}\; H_{1}: \boldsymbol{\mu}_1 \neq \boldsymbol{\mu}_2.$$
Zhang et al.(2020) proposed the following test statistic:
$$T_{ZGZC} = \frac{n_1n_2}{n} \|\bar{\boldsymbol{y}}_1 - \bar{\boldsymbol{y}}_2\|^2,$$
where \(\bar{\boldsymbol{y}}_{i},i=1,2\) are the sample mean vectors.
They showed that under the null hypothesis, \(T_{ZGZC}\) and a chi-squared-type mixture have the same normal or non-normal limiting distribution.