Consider $k$ independent random samples from $p$-dimensional multivariate normal distributions. We are interested in the limiting distribution of the log-likelihood ratio test statistics for testing for the equality of $k$ covariance matrices. It is well known from classical multivariate statistics that the limit is a chi-square distribution when $k$ and $p$ are fixed integers. Jiang and Yang~\cite{JY13} and Jiang and Qi~\cite{JQ15} have obtained the central limit theorem for the log-likelihood ratio test statistics when the dimensionality $p$ goes to infinity with the sample sizes. In this paper, we derive the central limit theorem when either $p$ or $k$ goes to infinity. We also propose adjusted test statistics which can be well approximated by chi-squared distributions regardless of values for $p$ and $k$. Furthermore, we present numerical simulation results to evaluate the performance of our adjusted test statistics and the log-likelihood ratio statistics based on classical chi-square approximation and the normal approximation.

翻译：考虑来自$p$维多元正态分布的$k$个独立随机样本。我们关注用于检验$k个$协方差矩阵相等性的对数似然比检验统计量的极限分布。经典多元统计理论中已知，当$k$和$p$为固定整数时，其极限为卡方分布。Jiang与Yang~\\cite {JY13}以及Jiang与Qi~\\cite {JQ15}已获得当维度$p$随样本量趋于无穷时对数似然比检验统计量的中心极限定理。本文推导了当$p$或$k$趋于无穷时的中心极限定理。同时，我们提出了一种调整后的检验统计量，该统计量无论$p$和$k$取值如何，均能很好地近似于卡方分布。此外，我们通过数值模拟结果评估了调整后检验统计量以及基于经典卡方近似和正态近似的对数似然比统计量的性能。