Minimax and Adaptive Inference in Nonparametric Function Estimation
TLDR
In this paper, the authors discuss minimaxity and adaptive minimaxness in nonparametric function estimation with respect to three interrelated problems, function estimation under global integrated squared error, estimation under pointwise squared error and non-parametric confidence intervals.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.read more
Citations
More filters
Posted Content
Minimax Estimation of Functionals of Discrete Distributions
TL;DR: The minimax rate-optimal mutual information estimator yielded by the framework leads to significant performance boosts over the Chow-Liu algorithm in learning graphical models and the practical advantages of the schemes for the estimation of entropy and mutual information.
Journal ArticleDOI
Minimax Estimation of Functionals of Discrete Distributions
TL;DR: In this article, a general methodology for the construction and analysis of essentially minimax estimators for a wide class of functionals of finite dimensional parameters, and elaborate on the case of discrete distributions, where the support size $S$ is unknown and may be comparable with or even much larger than the number of observations $n$.
Journal ArticleDOI
Minimax Estimation of Discrete Distributions Under $\ell _{1}$ Loss
TL;DR: In this article, the authors consider the problem of discrete distribution estimation under the assumption that the support size of the observations grows with the number of observations, and provide tight upper and lower bounds on the maximum risk of the empirical distribution.
Journal ArticleDOI
Asymptotically minimax empirical Bayes estimation of a sparse normal mean vector
Ryan Martin,Stephen G. Walker +1 more
TL;DR: A novel empirical Bayes model is proposed that admits a posterior distribution with desirable properties under mild conditions that concentrates on balls, centered at the true mean vector, with squared radius proportional to the minimax rate and its posterior mean is an asymptotically minimax estimator.
Journal ArticleDOI
Minimax estimation of linear functionals over nonconvex parameter spaces
T. Tony Cai,Mark G. Low +1 more
TL;DR: In this paper, the minimax theory for estimating linear functionals is extended to the case of a finite union of convex parameter spaces, and the results developed in this paper have important applications to the theory of adaptation.
References
More filters
Book
Ten lectures on wavelets
TL;DR: This paper presents a meta-analyses of the wavelet transforms of Coxeter’s inequality and its applications to multiresolutional analysis and orthonormal bases.
Journal ArticleDOI
Ten Lectures on Wavelets
TL;DR: In this article, the regularity of compactly supported wavelets and symmetry of wavelet bases are discussed. But the authors focus on the orthonormal bases of wavelets, rather than the continuous wavelet transform.
Journal ArticleDOI
Ideal spatial adaptation by wavelet shrinkage
TL;DR: In this article, the authors developed a spatially adaptive method, RiskShrink, which works by shrinkage of empirical wavelet coefficients, and achieved a performance within a factor log 2 n of the ideal performance of piecewise polynomial and variable-knot spline methods.
Journal ArticleDOI
Adapting to Unknown Smoothness via Wavelet Shrinkage
TL;DR: In this article, the authors proposed a smoothness adaptive thresholding procedure, called SureShrink, which is adaptive to the Stein unbiased estimate of risk (sure) for threshold estimates and is near minimax simultaneously over a whole interval of the Besov scale; the size of this interval depends on the choice of mother wavelet.
Book
Theory of function spaces
TL;DR: In this article, the authors measure smoothness using Atoms and Pointwise Multipliers, Wavelets, Spaces on Lipschitz Domains, Wavelet and Sampling Numbers.