# On the Fisher information matrix for multivariate elliptically contoured distributions

arXiv: https://arxiv.org/abs/1306.6415 · HTML: https://mlsanity.com/paper/1306.6415/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Jun 2013.

## Built on by (most cited first, 3 of 3)

- [The Fisher-Rao geometry of CES distributions](https://mlsanity.com/paper/2310.01032/) (2023)
- [FEMDA: a unified framework for discriminant analysis](https://mlsanity.com/paper/2311.07518/) (2023)
- [CRB-Driven Beamforming and Trajectory Optimization for UAV-assisted ISAC System](https://mlsanity.com/paper/2607.19609/) (2026)

## Abstract (arXiv)

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.