Testing the equality of two high-dimensional mean vectors is a fundamental problem in multivariate analysis. While the classical Hotelling's $T^2$ test is optimal in low-dimensional settings, it fails when the dimension $p$ is comparable to or exceeds the sample size $n$. Several extensions, including the Diagonal Likelihood Ratio Test (DLRT), have been proposed under the working independence assumption among variables. However, such an assumption can lead to a substantial loss of power when correlations are present. In this paper, we propose a new test, the Block Independent Likelihood Ratio Test (BILT), which generalizes DLRT by relaxing the working independence assumption to a block independence assumption. We establish its asymptotic normality of the null distribution of the BILT statistic for 'increasing $p$ with small $n$' under mild regularity conditions. We further analyze the asymptotic power of BILT under a local alternatives. Extensive simulation studies show that BILT maintains Type I error control and achieves substantially higher power than DLRT across a wide range of covariance structures. An application to the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset further demonstrates the application of BILT to testing mean differences between two matrix-variate populations.

翻译：检验两个高维均值向量的相等性是多元分析中的基本问题。经典 Hotelling $T^2$ 检验在低维环境下最优，但当维度 $p$ 与样本量 $n$ 相当或超过样本量时，该检验失效。基于变量间工作独立性假设，研究者已提出多种扩展方法，包括对角似然比检验（Diagonal Likelihood Ratio Test, DLRT）。然而，当变量间存在相关性时，此类假设可能导致检验功效显著下降。本文提出一种新型检验方法——块独立似然比检验（Block Independent Likelihood Ratio Test, BILT），该检验通过将工作独立性假设放宽为块独立性假设，推广了DLRT。在温和正则条件下，我们建立了"$p$ 递增而 $n$ 较小"情形下BILT统计量零分布的渐近正态性。进一步分析了局部备择假设下BILT的渐近功效。广泛模拟研究表明，在多种协方差结构下，BILT能有效控制第一类错误，且检验功效显著高于DLRT。基于阿尔茨海默病神经影像学倡议（ADNI）数据集的应用实例，进一步展示了BILT在两个矩阵变量总体均值差异检验中的实际效用。