The Semiparametric Cram\'er-Rao Bound for Complex Elliptically Symmetric Distributions

TL;DR

This paper extends the Semiparametric Cramér-Rao Bound to complex elliptically symmetric distributions, deriving a closed-form expression and illustrating its application to complex t-distributed vectors.

Contribution

It introduces the complex version of the CSCRB using Wirtinger calculus, expanding the theoretical framework for complex elliptically symmetric distributions.

Findings

Closed-form expression for CCSCRB derived

Application to complex t-distributed vectors demonstrated

Enhances theoretical tools for complex distribution estimation

Abstract

This letter aims at extending the Constrained Semiparametric Cramer-Rao Bound (CSCRB) for the joint estimation of mean vector and scatter matrix of Real Elliptically Symmetric (RES) distributions to Complex Elliptically Symmetric (CES) distributions. A closed form expression for the complex CSCRB (CCSCRB) is derived by exploiting the so-called \textit{Wirtinger} or $\mathbb{C}\mathbb{R}$-\textit{calculus}. Finally, the CCSCRB for the estimation of the complex mean vector and scatter matrix of a set of complex $t$-distributed random vectors is provided as an example of application.