In this paper, we discuss tests for mean vector of high-dimensional data when the dimension $p$ is a function of sample size $n$. One of the tests, called the decomposite $T^{2}$-test, in the high-dimensional testing problem is constructed based on the estimation work of Ledoit and Wolf (2018), which is an optimal orthogonally equivariant estimator of the inverse of population covariance matrix under Stein loss function. The asymptotic distribution function of the test statistic is investigated under a sequence of local alternatives. The asymptotic relative efficiency is used to see whether a test is optimal and to perform the power comparisons of tests. An application of the decomposite $T^{2}$-test is in testing significance for the effect of monthly unlimited transport policy on public transportation, in which the data are taken from Taipei Metro System.

翻译：本文探讨了当维度$p$为样本量$n$的函数时，高维数据均值向量的检验问题。其中一个检验方法——即高维检验问题中的分解$T^{2}$检验——是基于Ledoit和Wolf（2018）的估计工作构建的。该方法在Stein损失函数下，提供了总体协方差矩阵逆的最优正交等变估计。我们研究了在局部备择假设序列下检验统计量的渐近分布函数，并利用渐近相对效率判断检验的最优性，同时进行检验势的比较。分解$T^{2}$检验的一个应用案例是对台北捷运系统中月度无限次交通政策效果的显著性检验，所用数据来自台北捷运系统。