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October 2022 Relaxing the Gaussian assumption in shrinkage and SURE in high dimension
Max Fathi, Larry Goldstein, Gesine Reinert, Adrien Saumard
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Ann. Statist. 50(5): 2737-2766 (October 2022). DOI: 10.1214/22-AOS2208

Abstract

Shrinkage estimation is a fundamental tool of modern statistics, pioneered by Charles Stein upon his discovery of the famous paradox involving the multivariate Gaussian. A large portion of the subsequent literature only considers the efficiency of shrinkage, and that of an associated procedure known as Stein’s Unbiased Risk Estimate, or SURE, in the Gaussian setting of that original work. We investigate what extensions to the domain of validity of shrinkage and SURE can be made away from the Gaussian through the use of tools developed in the probabilistic area now known as Stein’s method. We show that shrinkage is efficient away from the Gaussian under very mild conditions on the distribution of the noise. SURE is also proved to be adaptive under similar assumptions, and in particular in a way that retains the classical asymptotics of Pinsker’s theorem. Notably, shrinkage and SURE are shown to be efficient under mild distributional assumptions, and particularly for general isotropic log-concave measures.

Funding Statement

MF was additionally supported by ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02, as well as Projects EFI (ANR-17-CE40-0030) and MESA (ANR-18-CE40-006) of the French National Research Agency (ANR).
GR is partially supported by EPSRC grants EP/R018472/1 and EP/T018445/1.

Acknowledgments

This research was started during the workshop Stein’s method and applications in high-dimensional statistics at the American Institute of Mathematics (AIM), San Jose, California, and we acknowledge the generous support and the fertile research environment provided by AIM. The authors sincerely thank the Associate Editor and our two reviewers for their hard work in providing us with detailed and insightful reviews. The fourth author warmly thanks Lionel Truquet for instructive discussions related to martingale difference random fields.

Citation

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Max Fathi. Larry Goldstein. Gesine Reinert. Adrien Saumard. "Relaxing the Gaussian assumption in shrinkage and SURE in high dimension." Ann. Statist. 50 (5) 2737 - 2766, October 2022. https://doi.org/10.1214/22-AOS2208

Information

Received: 1 April 2020; Revised: 1 March 2022; Published: October 2022
First available in Project Euclid: 27 October 2022

Digital Object Identifier: 10.1214/22-AOS2208

Subjects:
Primary: 62F12 , 62F35

Keywords: shrinkage estimation , Stein kernel , unbiased risk estimation , zero bias

Rights: Copyright © 2022 Institute of Mathematical Statistics

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Vol.50 • No. 5 • October 2022
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