Testing the equality of mean vectors across $g$ different groups plays an important role in many scientific fields. In regular frameworks, likelihood-based statistics under the normality assumption offer a general solution to this task. However, the accuracy of standard asymptotic results is not reliable when the dimension $p$ of the data is large relative to the sample size $n_i$ of each group. We propose here an exact directional test for the equality of $g$ normal mean vectors with identical unknown covariance matrix, provided that $\sum_{i=1}^g n_i \ge p+g+1$. In the case of two groups ($g=2$), the directional test is equivalent to the Hotelling's $T^2$ test. In the more general situation where the $g$ independent groups may have different unknown covariance matrices, although exactness does not hold, simulation studies show that the directional test is more accurate than most commonly used likelihood based solutions. Robustness of the directional approach and its competitors under deviation from multivariate normality is also numerically investigated.

翻译：检验$g$个不同组别均值向量是否相等在众多科学领域中具有重要作用。在常规框架下，基于正态性假设的似然统计量为该问题提供了通用解决方案。然而，当数据维度$p$相对于各组样本量$n_i$较大时，标准渐近结果的准确性并不可靠。本文针对$\sum_{i=1}^g n_i \ge p+g+1$的条件，提出了一种用于检验$g$个正态总体均值向量（具有相同未知协方差矩阵）相等性的精确方向性检验。当$g=2$时，该方向性检验等价于Hotelling $T^2$检验。在更一般的情形下，若$g$个独立组别可能具有不同的未知协方差矩阵，尽管检验不再具有精确性，但仿真研究表明方向性检验的准确性优于大多数常用的基于似然的解法。同时，本文还通过数值模拟研究了方向性方法及其竞争方法在偏离多元正态性时的稳健性。