Statistics Papers

Document Type

Journal Article

Date of this Version

2012

Publication Source

Statistical Science

Volume

27

Issue

1

Start Page

31

Last Page

50

DOI

10.1214/11-STS355

Abstract

Since Stein’s 1956 seminal paper, shrinkage has played a fundamental role in both parametric and nonparametric inference. This article discusses minimaxity and adaptive minimaxity in nonparametric function estimation. Three interrelated problems, function estimation under global integrated squared error, estimation under pointwise squared error, and nonparametric confidence intervals, are considered. Shrinkage is pivotal in the development of both the minimax theory and the adaptation theory.

While the three problems are closely connected and the minimax theories bear some similarities, the adaptation theories are strikingly different. For example, in a sharp contrast to adaptive point estimation, in many common settings there do not exist nonparametric confidence intervals that adapt to the unknown smoothness of the underlying function. A concise account of these theories is given. The connections as well as differences among these problems are discussed and illustrated through examples.

Copyright/Permission Statement

The original published work is available at: https://projecteuclid.org/euclid.ss/1331729981#abstract

Keywords

Adaptation, adaptive estimation, Bayes minimax, Besov ball, block thresholding, confidence interval, ellipsoid, information pooling, linear functional, linear minimaxity, minimax, nonparametric regression, oracle, separable rules, sequence model, shrinkage, thresholding, wavelet, white noise model

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Date Posted: 27 November 2017