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.

翻译：大多数信号处理和统计应用高度依赖于特定的数据分布模型。尽管高斯分布是最常见的选择，但在大多数现实场景中，由于无法处理来自重尾总体或被异常值污染的数据，高斯分布并不适用。此类问题需要采用稳健统计方法。稳健模型和估计量通常基于椭圆总体，这使得椭圆分布广泛存在于所有稳健统计方法中。为确定此类工具在特定情形下的适用性，需使用拟合优度检验来验证椭圆性假设。椭圆性拟合优度检验通常难以分析，且其统计功效往往不够强。本研究在假定真实协方差矩阵未知的情况下，设计并严格分析了一种稳健的拟合优度检验，该检验在单位球面上可一致拒绝所有非椭圆性的备择假设。所提出的检验基于Tyler估计量，并以易于计算的统计数据形式表述。为进行严格分析，我们基于de Finetti引入的可交换随机变量演算，开发了一个新颖的框架。通过数值模拟将我们的结果与其他流行的拟合优度检验进行比较，结果表明所提出的方法具有显著更高的统计功效。