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2014 Estimation of covariance and precision matrices under scale-invariant quadratic loss in high dimension
Tatsuya Kubokawa, Akira Inoue
Electron. J. Statist. 8(1): 130-158 (2014). DOI: 10.1214/14-EJS878


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.


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Tatsuya Kubokawa. Akira Inoue. "Estimation of covariance and precision matrices under scale-invariant quadratic loss in high dimension." Electron. J. Statist. 8 (1) 130 - 158, 2014.


Published: 2014
First available in Project Euclid: 10 February 2014

zbMATH: 1282.62140
MathSciNet: MR3165436
Digital Object Identifier: 10.1214/14-EJS878

Primary: 62F10 , 62H12
Secondary: 62J07

Keywords: asymptotic expansion , Covariance matrix , high dimension , Moore-Penrose inverse , multivariate normal distribution , point estimation , precision matrix , ridge estimator , risk comparison , scale-invariant quadratic loss , Stein-Haff identity , Wishart distribution

Rights: Copyright © 2014 The Institute of Mathematical Statistics and the Bernoulli Society


Vol.8 • No. 1 • 2014
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