This work is concerned with testing the proportionality between two high dimensional covariance matrices. Several tests for proportional covariance matrices, based on modifying the classical likelihood ratio test and applicable in high dimension, have been proposed in the literature. Despite their usefulness, they tend to have unsatisfactory performance for nonnormal high dimensional multivariate data in terms of size or power. This article proposes a new high dimensional test by developing a bias correction to the existing test statistic constructed based on a scaled distance measure. The suggested test is nonparametric without requiring any specific parametric distribution such as the normality assumption. It can accommodate scenarios where the data dimension $p$ is greater than the sample size $n$, namely the “large $p$, small $n$” problem. With the aid of tools in modern probability theory, we study theoretical properties of the newly proposed test, which include the asymptotic normality and a power evaluation. We demonstrate empirically that our proposal has good size and power performances for a range of dimensions, sample sizes and distributions in comparison with the existing counterparts.

这项工作与测试两个高维协方差矩阵之间的比例有关。文献中提出了几种基于修正经典似然比检验并适用于高维的比例协方差矩阵检验。尽管它们很有用，但它们在大小或功效方面对非正则高维多元数据的性能往往不尽人意。本文通过对基于比例距离度量构建的现有测试统计数据进行偏差校正来提出一种新的高维测试。建议的测试是非参数的，不需要任何特定的参数分布，例如正态性假设。它可以适应数据维度$p$ 大于样本量 $ñ$，即“大 $p$， 小的 $ñ$“ 问题。借助现代概率论中的工具，我们研究了新提出的检验的理论性质，包括渐近正态性和功效评估。我们凭经验证明，与现有的同类产品相比，我们的建议在各种尺寸，样本量和分布范围内均具有良好的尺寸和功率性能。