Testing the Diagonality of a Large Covariance Matrix in a Regression Setting
针对响应变量维度远大于样本量的情况，提出一个偏差校正检验，用于判断解释变量能否解释变量间的线性关系，通过检验残差协方差矩阵的非对角元素是否显著。
In multivariate analysis, the covariance matrix associated with a set of variables of interest (namely response variables) commonly contains valuable information about the dataset. When the dimension of response variables is considerably larger than the sample size, it is a nontrivial task to assess whether there are linear relationships between the variables. It is even more challenging to determine whether a set of explanatory variables can explain those relationships. To this end, we develop a bias-corrected test to examine the significance of the off-diagonal elements of the residual covariance matrix after adjusting for the contribution from explanatory variables. We show that the resulting test is asymptotically normal. Monte Carlo studies and a numerical example are presented to illustrate the performance of the proposed test.