Minimax and Adaptive Inference in Nonparametric Function Estimation

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.

Published in at http://dx.doi.org/10.1214/11-STS355 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (13)

• Empirical Bayes selection of wavelet thresholds
• A data-driven block thresholding approach to wavelet estimation
• An adaptation theory for nonparametric confidence intervals
• Adaptive nonparametric confidence sets
• Nonquadratic estimators of a quadratic functional
• Adaptive confidence balls
• Confidence sets for nonparametric wavelet regression
• General empirical Bayes wavelet methods and exactly adaptive minimax estimation
• Optimal adaptive estimation of a quadratic functional
• Equivalence theory for density estimation, Poisson processes and Gaussian white noise with drift
• Minimax estimation of linear functionals over nonconvex parameter spaces
• On adaptive estimation of linear functionals
• Adapting to Unknown Sparsity by controlling the False Discovery Rate

Cited by in corpus (10)

• Asymptotically minimax empirical Bayes estimation of a sparse normal mean vector
• Estimating Optimal Treatment Rules with an Instrumental Variable: A Partial Identification Learning Approach
• Adaptive Estimation of Shannon Entropy
• Improved minimax estimation of a multivariate normal mean under heteroscedasticity
• Minimax Estimation of Discrete Distributions under ${\ell }_1$ Loss
• Polynomial Cost of Adaptation for X -Armed Bandits
• Adaptive Inference in Multivariate Nonparametric Regression Models Under Monotonicity
• Statistical and Computational Efficiency for Smooth Tensor Estimation with Unknown Permutations
• Optimal Adaptive Inference in Random Design Binary Regression
• Nonparametric principal subspace regression