Relaxing the Gaussian assumption in shrinkage and SURE in high dimension – Annals of Statistics 

Co-authored by Max Fathi, Larry Goldstein, Gesine Reinert, and Adrien Saumard, associate professor of Statistics at ENSAI and researcher at CREST, the paper “Relaxing the Gaussian assumption in shrinkage and SURE in high dimension” was published in volume 50 of the Annals of Statistics 

The Annals of Statistics is a peer-reviewed Statistics journal published by the Institute of Mathematical Statistics 


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. 


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

Author Affiliations

Max Fathi: Université Paris Cité and Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions & Laboratoire de Probabilités, Statistique et Modélisation 

Larry Goldstein: Department of Mathematics, University of Southern California 

Gesine Reinert: Department of Statistics, University of Oxford 

Adrien Saumard: ENSAI, CREST-UMR 9194


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Find out more about Adrien Saumard and research at ENSAI.