Estimation of covariance and precision matrices under scale-invariant quadratic loss in high dimension

Abstract

The problem of estimating covariance and precision matrices of multivariate normal distributions is addressed when both the sample size and the dimension of variables are large. The estimation of the precision matrix is important in various statistical inference including the Fisher linear discriminant analysis, confidence region based on the Mahalanobis distance and others. A standard estimator is the inverse of the sample covariance matrix, but it may be instable or can not be defined in the high dimension. Although (adaptive) ridge type estimators are alternative procedures which are useful and stable for large dimension. However, we are faced with questions about how to choose ridge parameters and their estimators and how to set up asymptotic order in ridge functions in high dimensional cases. In this paper, we consider general types of ridge estimators for covariance and precision matrices, and derive asymptotic expansions of their risk functions. Then we suggest the ridge functions so that the second order terms of risks of ridge estimators are smaller than those of risks of the standard estimators.