Nuisance parameters and elliptically symmetric distributions: a geometric approach to parametric and semiparametric efficiency

TL;DR

This paper investigates the efficiency of parameter estimation in elliptically symmetric distributions, explicitly computing projection operators for finite and infinite-dimensional nuisance parameters, and extends results to complex cases.

Contribution

It provides explicit formulas for the projection operator in elliptic models without asymptotic approximations, enhancing understanding of semiparametric efficiency with finite and infinite-dimensional nuisances.

Findings

Explicit projection operator formulas for elliptic models.

Extension of results to complex elliptically symmetric distributions.

Application to low-rank parameterizations.

Abstract

Elliptically symmetric distributions are a classic example of a semiparametric model where the location vector and the scatter matrix (or a parameterization of them) are the two finite-dimensional parameters of interest, while the density generator represents an \textit{infinite-dimensional nuisance} term. This basic representation of the elliptic model can be made more accurate, rich, and flexible by considering additional \textit{finite-dimensional nuisance} parameters. Our aim is therefore to investigate the deep and counter-intuitive links between statistical efficiency in estimating the parameters of interest in the presence of both finite and infinite-dimensional nuisance parameters. Previous seminal works have addressed this problem by leveraging a general result: if the statistical model has a specific group invariance, then the projection operator onto the semiparametric nuisance tangent space can be asymptotically expressed as a conditional expectation with respect to the maximal invariant sub-$\sigma$ algebra. In this article, we show that, for the statistical model of elliptical distributions, the projection operator can be explicitly computed without relying on the above-mentioned asymptotic approximation. This allows us to obtain original results also for the case in which the location vector and the scatter matrix are parameterized by a finite-dimensional vector that can be partitioned in two sub-vectors: one containing the parameters of interest and the other containing the nuisance parameters. As an example, we illustrate how the obtained results can be applied to the well-known \virg{low-rank} parameterization. Furthermore, while the theoretical analysis will be developed for Real Elliptically Symmetric (RES) distributions, we show how to extend our results to the case of Circular and Non-Circular Complex Elliptically Symmetric (C-CES and NC-CES) distributions.