On the Fisher information matrix for multivariate elliptically contoured distributions

TL;DR

This paper extends the Slepian-Bangs formula to elliptically contoured distributions, providing a closed-form Fisher information matrix expression that simplifies derivations for a broad family of distributions beyond Gaussian.

Contribution

It derives a closed-form Fisher information matrix for elliptically contoured distributions, generalizing the Gaussian case and including corrective factors based on the modular variate.

Findings

The new FIM formula reduces to the Gaussian case when applicable.

Provides a practical example with Student distributions.

Simplifies FIM derivation for EC distributions.

Abstract

The Slepian-Bangs formula provides a very convenient way to compute the Fisher information matrix (FIM) for Gaussian distributed data. The aim of this letter is to extend it to a larger family of distributions, namely elliptically contoured (EC) distributions. More precisely, we derive a closed-form expression of the FIM in this case. This new expression involves the usual term of the Gaussian FIM plus some corrective factors that depend only on the expectations of some functions of the so-called modular variate. Hence, for most distributions in the EC family, derivation of the FIM from its Gaussian counterpart involves slight additional derivations. We show that the new formula reduces to the Slepian-Bangs formula in the Gaussian case and we provide an illustrative example with Student distributions on how it can be used.