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.

翻译：椭圆对称分布是半参数模型的一个经典范例，其中位置向量与散布矩阵（或其参数化形式）是两个有限维的兴趣参数，而密度生成函数则代表一个\textit{无限维冗余}项。通过引入额外的\textit{有限维冗余}参数，可以使椭圆模型的基本表述更为精确、丰富且灵活。因此，本文旨在探究当同时存在有限维与无限维冗余参数时，兴趣参数估计的统计效率之间深刻且反直觉的内在关联。与以往基于Le Cam渐近理论处理该问题的研究不同，本文采用纯几何的研究路径：我们将借助投影、切空间等工具，在相关希尔伯特空间中对效率进行分析。这使得我们能够获得关于位置向量与散布矩阵由有限维向量参数化情况下的原创性结果，该参数向量可划分为两个子向量：一个包含兴趣参数，另一个包含冗余参数。作为示例，我们阐述了如何将所得结果应用于著名的"低秩"参数化场景。此外，虽然理论分析将针对实椭圆对称分布展开，但我们同时展示了如何将结论推广至圆对称与非圆对称复椭圆对称分布的情形。