Covariance matrices play a major role in statistics, signal processing and machine learning applications. This paper focuses on the semiparametric covariance/scatter matrix estimation problem in elliptical distributions. The class of elliptical distributions can be seen as a semiparametric model where the finite-dimensional vector of interest is given by the location vector and by the (vectorized) covariance/scatter matrix, while the density generator represents an infinite-dimensional nuisance function. The main aim of this work is then to provide possible estimators of the finite-dimensional parameter vector able to reconcile the two dichotomic concepts of robustness and (semiparametric) efficiency. An RR-estimator satisfying these requirements has been recently proposed by Hallin, Oja and Paindaveine for real-valued elliptical data by exploiting the Le Cam's theory of one-step efficient estimators and the rank-based statistics. In this paper, we firstly recall the building blocks underlying the derivation of such real-valued RR-estimator, then its extension to complex-valued data is proposed. Moreover, through numerical simulations, its estimation performance and robustness to outliers are investigated in a finite-sample regime.

中文翻译：

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复杂椭圆对称分布中的鲁棒半参数有效估计器


协方差矩阵在统计、信号处理和机器学习应用中发挥着重要作用。本文重点研究椭圆分布中的半参数协方差/散点矩阵估计问题。椭圆分布类可以看作是半参数模型，其中感兴趣的有限维向量由位置向量和（向量化）协方差/散布矩阵给出，而密度生成器表示无限维干扰函数。这项工作的主要目的是提供有限维参数向量的可能估计器，能够协调鲁棒性和（半参数）效率这两个二分概念。 Hallin、Oja 和 Paindaveine 最近通过利用 Le Cam 的一步有效估计器理论和基于等级的统计，针对实值椭圆数据提出了一种满足这些要求的 RR 估计器。在本文中，我们首先回顾了这种实值 RR 估计器的推导基础构建模块，然后提出了其对复值数据的扩展。此外，通过数值模拟，在有限样本条件下研究了其估计性能和对异常值的鲁棒性。