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

TL;DR

This paper revisits semiparametric estimation theory, offering a new geometric perspective on the Semiparametric Cramér-Rao Bound, which enhances understanding of parameter estimation in models with infinite-dimensional nuisance functions.

Contribution

It introduces a novel geometric Hilbert space approach to semiparametric estimation theory and derives a new form of the Semiparametric Cramér-Rao Bound.

Findings

Provides a new geometric interpretation of semiparametric models

Derives a novel form of the SCRB using Hilbert space methods

Enhances understanding of efficient estimation in complex models

Abstract

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.