Hypothesis testing under uniform-block covariance structures.

A block covariance structure is widely observed across large-scale and high-dimensional datasets in diverse fields such as biology, medicine, engineering, economics, and finance. This pattern entails partitioning a covariance matrix into uniform blocks, where each block exhibits equal variances and covariances. The importance of uniform-block structures lies in their ubiquity, interpretability, and ability to accommodate high dimensionality and data missingness. Despite their prevalence, statistical hypothesis testing under uniform-block covariance structures remains largely unexplored, and unknown statistical properties limit their application in research. To address this gap, we develop a comprehensive framework for hypothesis tests of both covariance structures and mean vectors, leveraging a novel block Hadamard product representation of uniform-block matrices. Specifically, we derive closed-form likelihood ratio test statistics and information statistics, explicitly establishing their null distributions. Additionally, we perform simultaneous marginal mean tests under a procedure that controls the false discovery proportion (FDP). Extensive simulations validate the consistency between theoretical and empirical distributions of the test statistics, assess the performance of the proposed FDP control procedure, and evaluate the robustness of the test statistics against structural disruptions, missing data, and distributional misspecification. Lastly, we apply our methodology to hypothesis testing in a high-dimensional imaging dataset.