Minimax and Adaptive Inference in Nonparametric Function Estimation

TL;DR

This paper reviews minimax and adaptive inference methods in nonparametric function estimation, highlighting theoretical differences and limitations, especially regarding confidence intervals and unknown smoothness.

Contribution

It provides a comprehensive overview of minimaxity and adaptation theories in nonparametric estimation, emphasizing the contrasting behaviors across different inference problems.

Findings

Shrinkage is central to minimax and adaptation theories.

Adaptive confidence intervals often do not exist in common settings.

Differences between point estimation and confidence interval adaptation are highlighted.

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.