GLOBAL UNIFORM RISK BOUNDS FOR WAVELET DECONVOLUTION ESTIMATORS

• 成果类型: Article
• 署名作者: Lounici, Karim; Nickl, Richard
• 署名单位: University of Cambridge
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/10-AOS836
• 发表日期: 2011
• 页码: 201-231
• 关键词: kernel density estimators; nonparametric deconvolution; Concentration inequalities; confidence bands; sharp optimality; LIMIT-THEOREMS; Optimal Rates; CONVERGENCE; minimization; Consistency

摘要

We consider the statistical deconvolution problem where one observes n replications from the model Y = X + epsilon, where X is the unobserved random signal of interest and epsilon is an independent random error with distribution phi. Under weak assumptions on the decay of the Fourier transform of phi, we derive upper bounds for the finite-sample sup-norm risk of wavelet deconvolution density estimators f(n) for the density f of X, where f : R -> R is assumed to be bounded. We then derive lower bounds for the minimax sup-norm risk over Besov balls in this estimation problem and show that wavelet deconvolution density estimators attain these bounds. We further show that linear estimators adapt to the unknown smoothness of f if the Fourier transform of phi decays exponentially and that a corresponding result holds true for the hard thresholding wavelet estimator if phi decays polynomially. We also analyze the case where f is a supersmooth/analytic density. We finally show how our results and recent techniques from Rademacher processes can be applied to construct global confidence bands for the density f.