Suppose we have the following \(k\) independent high-dimensional samples:
$$
\boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma},i=1,\ldots,k.
$$
It is of interest to test the following one-way MANOVA problem:
$$H_0: \boldsymbol{\mu}_1=\cdots=\boldsymbol{\mu}_k, \quad \text { vs. }\; H_1: H_0 \;\operatorname{is \; not\; ture}.$$
Schott (2007) proposed the following test statistic:
$$
T_{S}=[\operatorname{tr}(\boldsymbol{H})/h-\operatorname{tr}(\boldsymbol{E})/e]/\sqrt{N-1},
$$
where \(\boldsymbol{H}=\sum_{i=1}^kn_i(\bar{\boldsymbol{y}}_i-\bar{\boldsymbol{y}})(\bar{\boldsymbol{y}}_i-\bar{\boldsymbol{y}})^\top\), \(\boldsymbol{E}=\sum_{i=1}^k\sum_{j=1}^{n_i}(\boldsymbol{y}_{ij}-\bar{\boldsymbol{y}}_{i})(\boldsymbol{y}_{ij}-\bar{\boldsymbol{y}}_{i})^\top\), \(h=k-1\), and \(e=N-k\), with \(N=n_1+\cdots+n_k\).
They showed that under the null hypothesis, \(T_{S}\) is asymptotically normally distributed.