Robust and Semiparametric Efficient Estimators in Elliptical Distributions

This paper focuses on the semiparametric covariance/scatter matrix estimation problem in elliptical distributions. The class of elliptical distributions can be seen as a semiparametric model where the finite-dimensional vector of interest is given by the mean vector and by the (vectorized) covariance/scatter matrix, while the density generator represents an infinite-dimensional nuisance function. The main aim of this work is then to provide possible estimators of the finite-dimensional parameter vector able to reconcile the two dichotomic concepts of robustness and (semiparametric) efficiency. An $R$-estimator satisfying these requirements has been recently proposed by Hallin, Oja and Paindaveine for real-valued elliptical data by exploiting the Le Cam's theory of Local Asymptotic Normality (LAN) and the rank-based statistics. In this paper, we firstly provide a survey about the building blocks needed to derive such a robust and semiparametric efficient $R$-estimator, then its extension to complex-valued data is proposed. Finally, through numerical simulations, its estimation performance is investigated in finite-sample regime by comparing its Mean Squared Error with the Semiparametric Cram\'er-Rao Bound in different scenarios.