Specification tests for covariance structures in high-dimensional statistical models

We consider testing the covariance structure in statistical models. We focus on developing such tests when the random vectors of interest are not directly observable and have to be derived via estimated models. Additionally, the covariance specification may involve extra nuisance parameters which also need to be estimated. In a generic additive model setting, we develop and investigate test statistics based on the maximum discrepancy measure calculated from the residuals. To approximate the distributions of the test statistics under the null hypothesis, new multiplier bootstrap procedures with dedicated adjustments that incorporate the model and nuisance parameter estimation errors are proposed. Our theoretical development elucidates the impact due to the estimation errors with high-dimensional data and demonstrates the validity of our tests. Simulations and real data examples confirm our theory and demonstrate the performance of the proposed tests.