On the strong convergence of the optimal linear shrinkage estimator for large dimensional covariance matrix

Journal article (2014)

Authors

Taras Bodnar Humboldt-Universitat zu Berlin

Arjun K. Gupta Bowling Green State University

N. Parolya Leibniz Universität

Affiliation

External organisation

Random matrix theory Covariance matrix estimation Large-dimensional asymptotics

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Published Date

01-11-2014

Language

English

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Abstract

In this work we construct an optimal linear shrinkage estimator for the covariance matrix in high dimensions. The recent results from the random matrix theory allow us to find the asymptotic deterministic equivalents of the optimal shrinkage intensities and estimate them consistently. The developed distribution-free estimators obey almost surely the smallest Frobenius loss over all linear shrinkage estimators for the covariance matrix. The case we consider includes the number of variables p→. ∞ and the sample size n→. ∞ so that p/. n→. c∈. (0, +. ∞). Additionally, we prove that the Frobenius norm of the sample covariance matrix tends almost surely to a deterministic quantity which can be consistently estimated.