Covariance matrices of random vectors contain information that is crucial for modelling. Certain structures and patterns of the covariances (or correlations) may be used to justify parametric models, e.g., autoregressive models. Until now, there have been only few approaches for testing such covariance structures systematically and in a unified way. In the present paper, we propose such a unified testing procedure, and we will exemplify the approach with a large variety of covariance structure models. This includes common structures such as diagonal matrices, Toeplitz matrices, and compound symmetry but also the more involved autoregressive matrices. We propose hypothesis tests for these structures, and we use bootstrap techniques for better small-sample approximation. The structures of the proposed tests invite for adaptations to other covariance patterns by choosing the hypothesis matrix appropriately. We prove their correctness for large sample sizes. The proposed methods require only weak assumptions. With the help of a simulation study, we assess the small sample properties of the tests. We also analyze a real data set to illustrate the application of the procedure.

翻译：随机向量的协方差矩阵包含对建模至关重要的信息。协方差（或相关性）的特定结构与模式可用于证明参数模型（例如自回归模型）的合理性。迄今为止，仅有少数方法能够系统且统一地检验此类协方差结构。本文提出一种统一的检验程序，并通过多种协方差结构模型进行实例验证，涵盖对角矩阵、托普利茨矩阵、复合对称性等常见结构，以及更复杂的自回归矩阵。我们针对这些结构提出假设检验，并采用bootstrap技术改善小样本近似效果。所提出的检验结构允许通过适当选择假设矩阵来适配其他协方差模式。我们证明了该方法在大样本下的有效性，且仅需较弱的假设条件。通过模拟研究评估了检验的小样本性质，并结合真实数据集分析展示了该程序的应用。