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

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.