Robust Semiparametric Efficient Estimators in Elliptical Distributions

Covariance matrices play a major role in statistics, signal processing and machine learning applications. This paper focuses on the $\mathit{\text{semiparametric}}$ 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 location 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 $\mathit{\text{robustness}}$ and (semiparametric) $\mathit{\text{efficiency}}$. 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 $\mathit{\text{one-step}}\mathit{\text{efficient}}\mathit{\text{estimators}}$ and the $\mathit{\text{rank-based}}\mathit{\text{statistics}}$. In this paper, we firstly recall the building blocks underlying the derivation of such real-valued $R$-estimator, then its extension to complex-valued data is proposed. Moreover, through numerical simulations, its estimation performance and robustness to outliers are investigated in a finite-sample regime.