A fresh look at the Semiparametric Cram\'{e}r-Rao Bound

This paper aims at providing a fresh look at semiparametric estimation theory and, in particular, at the Semiparametric Cram\'{e}r-Rao Bound (SCRB). Semiparametric models are characterized by a finite-dimensional parameter vector of interest and by an infinite-dimensional nuisance function that is often related to an unspecified functional form of the density of the noise underlying the observations. We summarize the main motivations and the intuitive concepts about semiparametric models. Then we provide a new look at the classical estimation theory based on a geometrical Hilbert space-based approach. Finally, the semiparametric version of the Cram\'{e}r-Rao Bound for the estimation of the finite-dimensional vector of the parameters of interest is provided.