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February 2012 Minimax and Adaptive Inference in Nonparametric Function Estimation
T. Tony Cai
Statist. Sci. 27(1): 31-50 (February 2012). DOI: 10.1214/11-STS355


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.


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T. Tony Cai. "Minimax and Adaptive Inference in Nonparametric Function Estimation." Statist. Sci. 27 (1) 31 - 50, February 2012.


Published: February 2012
First available in Project Euclid: 14 March 2012

zbMATH: 1330.62059
MathSciNet: MR2953494
Digital Object Identifier: 10.1214/11-STS355

Rights: Copyright © 2012 Institute of Mathematical Statistics


Vol.27 • No. 1 • February 2012
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