椭圆对称性的稳健检验
A Robust Test for Elliptical Symmetry
AI总结：
本文针对未知协方差矩阵情形，基于 Tyler 估计量和可交换随机变量演算框架，提出并严格分析了一种在单位球面上对所有非椭圆备择假设一致且统计功效显著更高的稳健椭圆性拟合优度检验。
摘要：
大多数信号处理和统计应用都高度依赖特定的数据分布模型。高斯分布虽然是最常见的选择，但在大多数现实场景中并不充分，因为它们无法解释来自重尾总体或被离群值污染的数据。此类问题需要运用稳健统计学。稳健模型和估计量通常基于椭圆总体，这使得后者在所有稳健统计方法中无处不在。为了确定这些工具在特定情况下是否适用，使用拟合优度（GoF）检验来验证椭圆性假设。椭圆性 GoF 检验通常难以分析，且其统计功效往往不是特别强。在本工作中，假设真实协方差矩阵未知，我们设计并严格分析了一种对单位球面上所有非椭圆备择假设均具有一致性的稳健 GoF 检验。所提出的检验基于 Tyler 估计量，并以易于计算的数据统计量形式表述。为了对其进行严格分析，我们开发了一个基于 de Finetti 引入的可交换随机变量演算的新框架。我们的发现得到了数值模拟的支持，将所提出的检验与其他流行的 GoF 检验进行比较，并证明了所建议技术具有显著更高的统计功效。
英文摘要：
Most signal processing and statistical applications heavily rely on specific data distribution models. The Gaussian distributions, although being the most common choice, are inadequate in most real world scenarios as they fail to account for data coming from heavy-tailed populations or contaminated by outliers. Such problems call for the use of Robust Statistics. The robust models and estimators are usually based on elliptical populations, making the latter ubiquitous in all methods of robust statistics. To determine whether such tools are applicable in any specific case, goodness-of-fit (GoF) tests are used to verify the ellipticity hypothesis. Ellipticity GoF tests are usually hard to analyze and often their statistical power is not particularly strong. In this work, assuming the true covariance matrix is unknown we design and rigorously analyze a robust GoF test consistent against all alternatives to ellipticity on the unit sphere. The proposed test is based on Tyler's estimator and is formulated in terms of easily computable statistics of the data. For its rigorous analysis, we develop a novel framework based on the exchangeable random variables calculus introduced by de Finetti. Our findings are supported by numerical simulations comparing them to other popular GoF tests and demonstrating the significantly higher statistical power of the suggested technique.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
