Covariance matrix estimation under data-based loss

TL;DR

This paper develops new covariance matrix estimators for multivariate regression models with elliptically symmetric distributions, demonstrating improved risk performance over traditional methods through theoretical analysis and numerical validation.

Contribution

It introduces alternative covariance estimators with lower risk under data-based loss for elliptically distributed data, expanding beyond classical estimators.

Findings

New estimators outperform traditional ones in risk

Broader class of distributions where improvements hold

Numerical results confirm theoretical advantages

Abstract

In this paper, we consider the problem of estimating the $p\times p$ scale matrix $\Sigma$ of a multivariate linear regression model $Y=X\,\beta + \mathcal{E}\,$ when the distribution of the observed matrix $Y$ belongs to a large class of elliptically symmetric distributions. After deriving the canonical form $(Z^T U^T)^T$ of this model, any estimator $\hat{ \Sigma}$ of $\Sigma$ is assessed through the data-based loss tr$(S^{+}\Sigma\, (\Sigma^{-1}\hat{\Sigma} - I_p)^2 )\,$ where $S=U^T U$ is the sample covariance matrix and $S^{+}$ is its Moore-Penrose inverse. We provide alternative estimators to the usual estimators $a\,S$, where $a$ is a positive constant, which present smaller associated risk. Compared to the usual quadratic loss tr$(\Sigma^{-1}\hat{\Sigma} - I_p)^2$, we obtain a larger class of estimators and a wider class of elliptical distributions for which such an improvement occurs. A numerical study illustrates the theory.