2025/10/15 by Lap Lau, Lau, Lap Chi, Akshay Ramachandran +1
Mathematics · #Advanced Statistical Methods and Models #Convergence (economics) #Covariance #Covariance matrix #Covariance operator #Elliptical distribution #Estimator #Gaussian #Iterative method #Scaling #Statistical Distribution Estimation and Applications #Statistical Methods and Inference
paper · pdf · doi:10.48550/arxiv.2510.13751
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2025/10/15 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/28
A fundamental problem in statistics is estimating the shape matrix of an Elliptical distribution. This generalizes the familiar problem of Gaussian covariance estimation, for which the sample covariance achieves optimal estimation error. For Elliptical distributions, Tyler proposed a natural M-estimator and showed strong statistical properties in the asymptotic regime, independent of the underlying distribution. Numerical experiments show that this estimator performs very well, and that Tyler's iterative procedure converges quickly to the estimator. Franks and Moitra recently provided the first distribution-free error bounds in the finite sample setting, as well as the first rigorous convergence analysis of Tyler's iterative procedure. However, their results exceed the sample complexity of the Gaussian setting by a log2 d factor. We close this gap by proving optimal sample threshold and error bounds for Tyler's M-estimator for all Elliptical distributions, fully matching the Gaussian result. Moreover, we recover the algorithmic convergence even at this lower sample threshold. Our approach builds on the operator scaling connection of Franks and Moitra by introducing a novel pseudorandom condition, which we call ∞-expansion. We show that Elliptical distributions satisfy ∞-expansion at the optimal sample threshold, and then prove a novel scaling result for inputs satisfying this condition.