发表机构
Whiting School of Engineering, Johns Hopkins University; Department of Applied Mathematics and Statistics, Johns Hopkins University; Department of Statistics & Data Science Institute, University of Chicago（约翰斯·霍普金斯大学惠廷工程学院; 约翰斯·霍普金斯大学应用数学与统计学系; 芝加哥大学统计与数据科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对重尾分布下的稳健形状估计，证明空间符号协方差矩阵估计量（SSCM）在高维比例渐近下通过隐式正则化在Frobenius风险上优于Tyler的M估计量，并通过数值实验验证。
AI 中文摘要
协方差估计是系统辨识和数据驱动控制中许多应用的关键组成部分。尽管重尾分布可能缺乏可估计的协方差矩阵，但形状矩阵为广泛的椭圆分布族提供了一个定义明确、尺度自由的推广。在此背景下，从业者常采用Tyler的M估计量（TME），该估计量隐式定义并通过定点迭代计算。空间符号协方差矩阵估计量（SSCM）提供了一种更简单的替代方案：它对应于TME的一次迭代。然而，由于在固定维度渐近下统计不一致，SSCM在很大程度上被视为较差的估计量。相比之下，利用随机矩阵理论的二阶工具，我们证明在标准假设下，当维度和样本量按比例增长时，SSCM在Frobenius风险下渐近优于TME。这一优势源于隐式正则化，它以随维度增长而消失的偏差为代价降低了方差。数值实验即使在中等维度下也支持理论预测。
英文摘要
Covariance estimation is a key component of many applications in system identification and data-driven control. Although heavy-tailed distributions may lack a covariance matrix to estimate, the shape matrix provides a well-defined, scale-free generalization for the broad family of elliptical distributions. In this setting, practitioners often employ Tyler's M-estimator (TME), which is defined implicitly and is computed using a fixed-point iteration. The spatial sign covariance matrix estimator (SSCM) offers a much simpler alternative: it corresponds to one iteration of TME. Yet, the SSCM has been largely regarded as an inferior estimator due to its statistical inconsistency under fixed-dimensional asymptotics. By contrast, using second-order tools from random matrix theory, we establish that, under standard assumptions, SSCM asymptotically dominates TME in Frobenius risk when dimension and sample size grow proportionally. This advantage arises from implicit regularization, which reduces variance at the cost of a bias that vanishes as the dimension grows. Numerical experiments support the theoretical predictions even at moderate dimensions.
Comments19 pages, 3 figures
