Semiparametric inference and lower bounds for real elliptically symmetric distributions

This paper has a twofold goal. The first aim is to provide a deeper understanding of the family of the real elliptically symmetric (RES) distributions by investigating their intrinsic semiparametric nature. The second aim is to derive a semiparametric lower bound for the estimation of the parametric component of the model. The RES distributions represent a semiparametric model, where the parametric part is given by the mean vector and by the scatter matrix, while the non-parametric, infinite-dimensional, part is represented by the density generator. Since, in practical applications, we are often interested only in the estimation of the parametric component, the density generator can be considered as nuisance. The first part of this paper is dedicated to conveniently place the RES distributions in the framework of the semiparametric group models. In the second part, building on the mathematical tools previously introduced, the constrained semiparametric Cramér-Rao bound (CSCRB) for the estimation of the mean vector and of the constrained scatter matrix of a RES distributed random vector is introduced. The CSCRB provides a lower bound on the mean squared error of any robust $M$-estimator of mean vector and scatter matrix when no a priori information on the density generator is available. A closed-form expression for the CSCRB is derived. Finally, in simulations, we assess the statistical efficiency of the Tyler's and Huber's scatter matrix $M$-estimators with respect to the CSCRB.