Minimax and Adaptive Inference in Nonparametric Function Estimation
Tony Cai
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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.
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