标题： 干扰参数和椭圆对称分布：参数和半参数效率的几何方法 显示英文标题

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

摘要： 椭圆对称分布是半参数模型的经典示例，其中位置向量和散度矩阵（或其参数化）是两个感兴趣的有限维参数，而密度生成器则表示一个\textit{无限维干扰}项。通过考虑额外的\textit{有限维干扰}参数，可以使椭圆模型的这种基本表示更加准确、丰富和灵活。因此，我们的目标是探究在存在有限维和无限维干扰参数的情况下，估计感兴趣参数的统计效率之间的深层且违反直觉的联系。与先前使用勒康渐近理论解决这个问题的研究不同，我们这里的方法纯粹是几何的：我们将使用诸如投影和嵌入相关希尔伯特空间的切线空间等工具来分析效率。这使我们能够在位置向量和散射矩阵由有限维向量参数化的情况下获得原始结果，该向量可以分为两个子向量：一个包含目标参数，另一个包含干扰参数。作为示例，我们说明了如何将所得结果应用于众所周知的\virg{低秩}参数化。此外，虽然理论分析将针对实椭圆对称 (RES) 分布进行，但我们将展示如何将我们的结果扩展到圆形和非圆形复椭圆对称 (C-CES 和 NC-CES) 分布的情况。 显示英文摘要

摘要： 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. Unlike previous works that addressed this problem using Le Cam's asymptotic theory, our approach here is purely geometric: efficiency will be analyzed using tools such as projections and tangent spaces embedded in the relevant Hilbert space. This allows us to obtain original results also for the case where 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.