Statistical Science

Minimax and Adaptive Inference in Nonparametric Function Estimation

T. Tony Cai

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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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Statist. Sci., Volume 27, Number 1 (2012), 31-50.

First available in Project Euclid: 14 March 2012

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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


Cai, T. Tony. Minimax and Adaptive Inference in Nonparametric Function Estimation. Statist. Sci. 27 (2012), no. 1, 31--50. doi:10.1214/11-STS355.

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