Asymptotic non-linear shrinkage and eigenvector overlap for weighted sample covariance

TL;DR

This paper derives asymptotic formulas for non-linear shrinkage of weighted sample covariance matrices and eigenvector overlaps, providing algorithms and testing robustness against heavy-tailed distributions.

Contribution

It introduces explicit formulas and algorithms for asymptotic non-linear shrinkage and eigenvector overlap estimation in weighted covariance matrices, extending previous work.

Findings

Effective non-linear shrinkage estimators demonstrated

Algorithms enable practical computation of formulas

Robustness confirmed for heavy-tailed distributions

Abstract

We compute asymptotic non-linear shrinkage formulas for covariance and precision matrix estimators for weighted sample covariances, and the joint sample-population eigenvector overlap distribution, in the spirit of Ledoit and P\'ech\'e. We detail explicitly the formulas for exponentially-weighted sample covariances. We propose an algorithm to numerically compute those formulas. Experimentally, we show the performance of the asymptotic non-linear shrinkage estimators. Finally, we test the robustness of the theory to a heavy-tailed distributions.