Open Access
October 2020 Analytical nonlinear shrinkage of large-dimensional covariance matrices
Olivier Ledoit, Michael Wolf
Ann. Statist. 48(5): 3043-3065 (October 2020). DOI: 10.1214/19-AOS1921


This paper establishes the first analytical formula for nonlinear shrinkage estimation of large-dimensional covariance matrices. We achieve this by identifying and mathematically exploiting a deep connection between nonlinear shrinkage and nonparametric estimation of the Hilbert transform of the sample spectral density. Previous nonlinear shrinkage methods were of numerical nature: QuEST requires numerical inversion of a complex equation from random matrix theory whereas NERCOME is based on a sample-splitting scheme. The new analytical method is more elegant and also has more potential to accommodate future variations or extensions. Immediate benefits are (i) that it is typically 1000 times faster with basically the same accuracy as QuEST and (ii) that it accommodates covariance matrices of dimension up to 10,000 and more. The difficult case where the matrix dimension exceeds the sample size is also covered.


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Olivier Ledoit. Michael Wolf. "Analytical nonlinear shrinkage of large-dimensional covariance matrices." Ann. Statist. 48 (5) 3043 - 3065, October 2020.


Received: 1 November 2018; Revised: 1 October 2019; Published: October 2020
First available in Project Euclid: 19 September 2020

MathSciNet: MR4152634
Digital Object Identifier: 10.1214/19-AOS1921

Primary: 62H12
Secondary: 15A52 , 62G20

Keywords: Hilbert transform , Kernel estimation , rotation equivariance

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 5 • October 2020
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