Estimations for some functions of covariance matrix in high dimension under non-normality and its applications

When we consider a statistical test in the high dimensional case, we often need estimators of the functions of the covariance matrix ΣΣ. Especially, it is needed to estimate a2=(1/p)trΣ2a2=(1/p)trΣ2. The unbiased and consistent estimator of a2a2 is proposed in preceding study when the population distribution is multivariate normal. But it is difficult to estimate in the non-normal case. So we propose the unbiased and consistent estimators for some functions of covariance matrix including a2a2 under the non-normal case. Through the numerical simulation, we confirmed the accuracy of the approximation of our proposed estimators. Using proposed estimators, we proposed a test for assessing multivariate normality of the high-dimensional data.