Due to their parsimony, separable covariance models have been popular in modeling matrix-variate data. However, the inference from such a model may be misleading if the population covariance matrix $Σ$ is actually non-separable, motivating the use of statistical tests of separability. The existing separability tests suffer mainly from two issues: 1) test statistics that are not well-defined in high-dimensional settings, 2) low power for small sample sizes and null distributions that depend on unknown parameters, preventing exact error rate control. To address these issues, we propose novel invariant tests using the core covariance matrix, a complementary notion to a separable covariance matrix. We show that testing separability of $Σ$ is equivalent to testing sphericity of its core component. With this insight, we construct test statistics that are well-defined in high-dimensional settings and have distributions that are invariant under the null hypothesis of separability, allowing for exact simulation of null distributions. We establish the asymptotic properties of some test statistics by proving the asymptotic spectral equivalence between the sample covariance matrix and its core in a $p/n\rightarrowγ\in(0,\infty)$ regime. The large power of our proposed tests relative to existing procedures is demonstrated numerically.

翻译：由于其简洁性，可分离协方差模型在矩阵型数据建模中广受欢迎。然而，若总体协方差矩阵 $Σ$ 实际不可分离，则基于此类模型的推断可能产生误导，这推动了可分离性统计检验方法的发展。现有可分离性检验主要存在两个问题：1）检验统计量在高维场景下定义不明确；2）小样本量下统计功效低，且零分布依赖于未知参数，导致无法精确控制错误率。为解决这些问题，我们利用核心协方差矩阵（可分离协方差矩阵的互补概念）提出新型不变检验。研究表明，$Σ$ 的可分离性检验等价于其核心分量的球面性检验。基于这一发现，我们构建了高维场景下定义明确的检验统计量，其分布在可分离性原假设下具有不变性，从而可精确模拟零分布。通过证明在 $p/n\rightarrowγ\in(0,\infty)$ 条件下样本协方差矩阵与其核心矩阵的渐近谱等价性，我们确立了部分检验统计量的渐近性质。数值实验表明，相较于现有方法，所提检验方法具有显著更高的统计功效。