Semiparametric Estimation in Elliptical Distributions

This chapter has the twofold aim of introducing in an intuitive and accessible manner the general framework of semiparametric inference and then of showing how it can be fruitfully applied to the joint estimation of the location vector and the covariance (or scatter) matrix of a set of elliptically distributed observations in the presence of an unknown density generator. A semiparametric model is a set of probablity density functions (pdfs) parameterized by a finite-dimensional parameter vector of interest and by an infinite-dimensional nuisance parameter, i.e., a function, whose estimation is not strictly required. The presence of this additional functional unknown will clearly lead to some performance losses in the estimation of the finite-dimensional parameter vector of interest. The first goal of this chapter is then to show how the classical estimation theory can be generalized in order to take into account an infinite-dimensional nuisance term. In particular, the three building blocks of the semiparametric theory, that are the Hilbert space of score vectors, the nuisance tangent space and the related projection operator, will be introduced. By means of these abstract concepts, we define the semiparametric counterpart of the Fisher Information Matrix (FIM) and the related semiparametric efficiency bound. After having prepared the theoretical ground, the focus of the second part of the chapter is on the application of the general semiparametric inference framework to the joint estimation of the location vector and of the scatter matrix in Real Elliptically Symmetric (RES) distributed random vectors. A closed-form expression for the semiparametric FIM and the related bound will be provided. We conclude the chapter by presenting the class of the R-estimators of the scatter matrix.