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.
Highlights
Statistical inference with high dimension has received much attention in recent years and has been actively studied from both theoretical and practical aspects in the literature
We evaluate the difference of risk functions of the ridge-type and the standard estimators asymptotically for large n and p, where the risk functions are measured with respect to the scale-invariant quadratic loss functions
We have considered estimation of the covariance and precision matrices by the ridge-type estimators, and have derived asymptotic expansions of their risk functions relative to the scale-invariant quadratic loss functions when the sample size and the dimension are very large
Summary
Statistical inference with high dimension has received much attention in recent years and has been actively studied from both theoretical and practical aspects in the literature. In the case of p > N , the inverse of the sample covariance matrix cannot be defined, and an estimator based on the Moore-Penrose generalized inverse of the sample covariance matrix has been used in Srivastava [11]. Another useful and stable estimator for the precision matrix is a ridge estimator, and its various variants have been used in literature. We address the problems of estimating both covariance matrix Σ and precision matrix Σ−1, and consider general ridge-type estimators, respectively given by ΣΛ = c(V + dΛ), and ΣΛ. Some technical tools and proofs are given in the appendix
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