  Covariance matrices play a major role in statistics, signal processing and
machine learning applications. This paper focuses on the
\textit{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 \textit{robustness} and (semiparametric) \textit{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 \textit{one-step efficient estimators} and the
\textit{rank-based 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.
