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This paper has a twofold goal. The first aim is to provide a deeperunderstanding of the family of the Real Elliptically Symmetric (RES)distributions by investigating their intrinsic semiparametric nature. Thesecond aim is to derive a semiparametric lower bound for the estimation of theparametric component of the model. The RES distributions represent asemiparametric model where the parametric part is given by the mean vector andby the scatter matrix while the non-parametric, infinite-dimensional, part isrepresented by the density generator. Since, in practical applications, we areoften interested only in the estimation of the parametric component, thedensity generator can be considered as nuisance. The first part of the paper isdedicated to conveniently place the RES distributions in the framework of thesemiparametric group models. The second part of the paper, building on themathematical tools previously introduced, the Constrained SemiparametricCram\'{e}r-Rao Bound (CSCRB) for the estimation of the mean vector and of theconstrained scatter matrix of a RES distributed random vector is introduced.The CSCRB provides a lower bound on the Mean Squared Error (MSE) of any robust$M$-estimator of mean vector and scatter matrix when no a-priori information onthe density generator is available. A closed form expression for the CSCRB isderived. Finally, in simulations, we assess the statistical efficiency of theTyler's and Huber's scatter matrix $M$-estimators with respect to the CSCRB.

Semiparametric Inference and Lower Bounds for Real Elliptically Symmetric Distributions