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

Taras Bodnar, Arjun K. Gupta, Nestor Parolya

Research output: Contribution to journalArticleScientificpeer-review

16 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)215-228
Number of pages14
JournalJournal of Multivariate Analysis
Volume132
DOIs
Publication statusPublished - 1 Nov 2014
Externally publishedYes

Keywords

  • Covariance matrix estimation
  • Large-dimensional asymptotics
  • Random matrix theory

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