  This paper aims at presenting a simulative analysis of the main properties of
a new $R$-estimator of shape matrices in Complex Elliptically Symmetric (CES)
distributed observations. First proposed by Hallin, Oja and Paindaveine for the
real-valued case and then extended to the complex field in our recent work,
this $R$-estimator has the remarkable property to be, at the same time,
\textit{distributionally robust} and \textit{semiparametric efficient}. Here,
the efficiency of different possible configurations of this $R$-estimator are
investigated by comparing the resulting Mean Square Error (MSE) with the
Constrained Semiparametric Cram\'{e}r-Rao Bound (CSCRB). Moreover, its
robustness to outliers is assessed and compared with the one of the celebrated
Tyler's estimator.
