  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.

Methodology