Shrinkage-to-tapering estimation of large covariance matrices

Xiaohui Chen, Z. Jane Wang, Martin J. McKeown

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper, we introduce a shrinkage-to-tapering approach for estimating large covariance matrices when the number of samples is substantially fewer than the number of variables (i.e., n,p → ∞ and p\n → ∞). The proposed estimator improves upon both shrinkage and tapering estimators by shrinking the sample covariance matrix to its tapered version. We first show that, under both normalized Frobenius and spectral risks, the minimum mean-squared error (MMSE) shrinkage-to-identity estimator is inconsistent and outperformed by a minimax tapering estimator for a class of high-dimensional and diagonally dominant covariance matrices. Motivated by this observation, we propose a shrinkage-to-tapering oracle (STO) estimator for efficient estimation of general, large covariance matrices. A closed-form formula of the optimal coefficient ρ of the proposed STO estimator is derived under the minimum Frobenius risk. Since the true covariance matrix is to be estimated, we further propose a STO approximating (STOA) algorithm with a data-driven bandwidth selection procedure to iteratively estimate the coefficient ρ and the covariance matrix. We study the finite sample performances of different estimators and our simulation results clearly show the improved performances of the proposed STO estimators. Finally, the proposed STOA method is applied to a real breast cancer gene expression data set.

Original languageEnglish (US)
Article number6252067
Pages (from-to)5640-5656
Number of pages17
JournalIEEE Transactions on Signal Processing
Volume60
Issue number11
DOIs
StatePublished - 2012
Externally publishedYes

Keywords

  • Large covariance estimation
  • minimax risk
  • minimum mean-squared errors
  • shrinkage estimator
  • tapering operator

ASJC Scopus subject areas

  • Signal Processing
  • Electrical and Electronic Engineering

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