Relaxing the Gaussian assumption in Shrinkage and SURE in high dimension
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Shrinkage estimation is a fundamental tool of modern statistics, pioneered by Charles Stein upon the discovery of his famous paradox. Despite a large subsequent literature, the efficiency of shrinkage, and the associated procedure known as Stein's Unbiased Risk Estimate, or SURE, has mainly been analysed in the Gaussian setting. Importing tools developed for use in the probabilistic area now known as Stein's method, the present work investigates the domain of validity of shrinkage and SURE away from the Gaussian. 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.
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