Optimal shrinkage covariance matrix estimation under random sampling from elliptical distributions

Esa Ollila, Elias Raninen

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28 Citations (Scopus)


This paper considers the problem of estimating a high-dimensional covariance matrix in a low sample support situation where the sample size is smaller, or not much larger, than the dimensionality of the data, which could potentially be very large. We develop a regularized sample covariance matrix (RSCM) estimator which can be applied in commonly occurring high-dimensional data problems. The proposed RSCM estimator is based on estimators of the unknown optimal (oracle) shrinkage parameters that yield the minimum mean squared error between the RSCM and the true covariance matrix when the data is sampled from an unspecified elliptically symmetric distribution. We propose two variants of the RSCM estimator which differ in the approach in which they estimate the underlying sphericity parameter involved in the theoretical optimal shrinkage parameter. The performance of the proposed RSCM estimators are evaluated with numerical simulation studies. In particular, when the sample sizes are low, the proposed RSCM estimators often show a significant improvement over the conventional RSCM estimator by Ledoit and Wolf (2004). We further evaluate the performance of the proposed estimators in a portfolio optimization problem with real data wherein the proposed methods are able to outperform the benchmark methods.

Original languageEnglish
Article number8676095
Pages (from-to)2707-2719
Number of pages13
JournalIEEE Transactions on Signal Processing
Issue number10
Early online date2019
Publication statusPublished - 15 May 2019
MoE publication typeA1 Journal article-refereed


  • Covariance matrices
  • Symmetric matrices
  • Estimation
  • Eigenvalues and eigenfunctions
  • Portfolios
  • Benchmark testing
  • Indexes
  • Sample covariance matrix
  • shrinkage estimation
  • regularization
  • elliptical distribution


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  • Robust Statistics for High-dimensional Data

    Ollila, E., Raninen, E., Basiri, S., Tabassum, M. N. & Mian, A.


    Project: Academy of Finland: Other research funding

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