We investigate the likelihood ratio test for a large block-diagonal covariance matrix with an increasing number of blocks under the null hypothesis. While so far the likelihood ratio statistic has only been studied for normal populations, we establish that its asymptotic behavior is invariant under a much larger class of distributions. This implies robustness against model misspecification, which is common in high-dimensional regimes. Demonstrating the flexibility of our approach, we additionally establish asymptotic normality of the log-likelihood ratio test statistic for the equality of many large sample covariance matrices under model uncertainty. For this statistic, a subtle adjustment to the centering term is needed compared to normal case. A simulation study and an analysis of a data set from psychology emphasize the usefulness of our findings.

翻译：我们研究了在原假设下具有递增块数的大块对角协方差矩阵的似然比检验。尽管迄今为止似然比统计量仅在正态总体中被研究，我们证明其渐近行为在更广泛的分布类别下保持不变。这意味着对模型误设具有稳健性，这在高维情形中十分常见。为展示我们方法的灵活性，我们还建立了模型不确定性下多个大样本协方差矩阵相等性的对数似然比检验统计量的渐近正态性。对于该统计量，与正态情形相比，其中心项需要微妙的调整。模拟研究和一项心理学数据集的分析强调了本研究结果的实际价值。