🤖 AI Summary
This paper addresses covariance and precision matrix estimation from weighted sample covariance matrices—particularly exponentially weighted ones—and proposes an asymptotically optimal nonlinear shrinkage method while characterizing the joint distribution of sample and population eigenvector overlaps. Leveraging random matrix theory and the Ledoit–Péché asymptotic framework, we derive, for the first time, explicit asymptotic formulas for nonlinear shrinkage under general weighting schemes; establish a rigorous theory for the joint distribution of eigenvector overlaps; and develop a computationally tractable, robust numerical algorithm. Experiments demonstrate substantial improvements over conventional linear shrinkage estimators. Crucially, both theoretical guarantees and algorithmic performance remain valid under heavy-tailed data distributions. Our core contributions are: (1) the first explicit asymptotic solution for weighted nonlinear shrinkage; (2) a precise characterization of the joint distribution of sample–population eigenvector overlaps; and (3) a unified estimation framework that achieves theoretical optimality while maintaining practical robustness.
📝 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.