Optimal shrinkage covariance matrix estimation under random sampling from elliptical distributions

TL;DR

This paper introduces a regularized covariance matrix estimator optimized for high-dimensional data from elliptical distributions, improving accuracy especially with small sample sizes in practical applications.

Contribution

It develops a novel RSCM estimator with data-driven shrinkage parameters tailored for elliptical distributions, outperforming existing methods in simulations and real-world tasks.

Findings

Proposed RSCM estimators outperform Ledoit-Wolf in low-sample regimes.

The estimators improve classification accuracy.

They enhance portfolio optimization results.

Abstract

This paper considers the problem of estimating a high-dimensional (HD) covariance matrix when 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 sparse data problems. The proposed RSCM estimator is based on estimators of the unknown optimal (oracle) shrinkage parameters that yield the minimum mean squared error (MMSE) 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 classification and portfolio optimization problems with real data wherein the proposed methods are able to outperform the benchmark methods.