DECONVOLUTION WITH UNKNOWN ERROR DISTRIBUTION

• 成果类型: Article
• 署名作者: Johannes, Jan
• 署名单位: Ruprecht Karls University Heidelberg
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/08-AOS652
• 发表日期: 2009
• 页码: 2301-2323
• 关键词: nonparametric deconvolution; tikhonov regularization; DENSITY-ESTIMATION; Optimal Rates; CONVERGENCE; estimator; models

摘要

We consider the problem of estimating a density f(X) using a sample Y-l,....Y-n from f(Y) = f(X) star f(is an element of), where f(is an element of) is an unknown density. We assume that all additional sample is an element of(l),...,is an element of(m) from f(is an element of) is observed. Estimators of f(X) and its derivatives are constructed by using nonparametric estimators of f(X) and f(is an element of) and by applying a spectral cut-off in the Fourier domain. We derive the rate of convergence of the estimators ill case of a known and unknown error density Where it is assumed that f(X) satisfies a polynomial, logarithmic or general source condition. It is shown that the proposed estimators are asymptotically optimal ill a minimax sense ill the models with known or unknown error density, if the density f(X) belongs to a Sobolev space H-p and f(is an element of) is ordinary smooth or supersmooth.