On the variability of the sample covariance matrix under complex elliptical distributions

TL;DR

This paper derives the variance-covariance matrix of the sample covariance matrix for complex elliptical distributions, providing theoretical insights and formulas for its mean squared error, with illustrative examples.

Contribution

It introduces a general formula for the variance-covariance matrix of affine equivariant matrix statistics under complex elliptical distributions, including the sample covariance matrix.

Findings

Derived the variance-covariance matrix for the sample covariance matrix.

Provided formulas for the mean squared error of the sample covariance matrix.

Presented illustrative examples demonstrating the formulas.

Abstract

We derive the form of the variance-covariance matrix for any affine equivariant matrix-valued statistics when sampling from complex elliptical distributions. We then use this result to derive the variance-covariance matrix of the sample covariance matrix (SCM) as well as its theoretical mean squared error (MSE) when finite fourth-order moments exist. Finally, illustrative examples of the formulas are presented.