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

arXiv: https://arxiv.org/abs/2506.23213 · HTML: https://mlsanity.com/paper/2506.23213/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jun 2025.

## Built on by (most cited first, 1 of 1)

- [Toward an Asymptotic Efficiency Theory on Regular Parameter Manifolds](https://mlsanity.com/paper/2510.13703/) (2025)

## Abstract (arXiv)

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 textitinfinite-dimensional nuisance term. This basic representation of the elliptic model can be made more accurate, rich, and flexible by considering additional textitfinite-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-σ 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 virglow-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.