  The joint estimation of the location vector and the shape matrix of a set of
independent and identically Complex Elliptically Symmetric (CES) distributed
observations is investigated from both the theoretical and computational
viewpoints. This joint estimation problem is framed in the original context of
semiparametric models allowing us to handle the (generally unknown) density
generator as an \textit{infinite-dimensional} nuisance parameter. In the first
part of the paper, a computationally efficient and memory saving implementation
of the robust and semiparmaetric efficient $R$-estimator for shape matrices is
derived. Building upon this result, in the second part, a joint estimator,
relying on the Tyler's $M$-estimator of location and on the $R$-estimator of
shape matrix, is proposed and its Mean Squared Error (MSE) performance compared
with the Semiparametric Cram\'{e}r-Rao Bound (CSCRB).
