Linear shrinkage of sample covariance matrix or matrices under elliptical distributions: a review

TL;DR

This review discusses linear shrinkage methods for sample covariance matrices under elliptical distributions, highlighting optimal coefficients based on key distribution parameters and applications in portfolio optimization.

Contribution

It provides a comprehensive overview of shrinkage estimators under elliptical distributions, including derivation of optimal coefficients and their application in finance.

Findings

Optimal shrinkage coefficients depend on elliptical kurtosis and sphericity.

Shrinkage estimators improve covariance matrix estimation accuracy.

Applications demonstrated in portfolio optimization.

Abstract

This chapter reviews methods for linear shrinkage of the sample covariance matrix (SCM) and matrices (SCM-s) under elliptical distributions in single and multiple populations settings, respectively. In the single sample setting a popular linear shrinkage estimator is defined as a linear combination of the sample covariance matrix (SCM) with a scaled identity matrix. The optimal shrinkage coefficients minimizing the mean squared error (MSE) under elliptical sampling are shown to be functions of few key parameters only, such as elliptical kurtosis and sphericity parameter. Similar results and estimators are derived for multiple population setting and applications of the studied shrinkage estimators are illustrated in portfolio optimization.