On the Optimal Rates of Convergence for Nonparametric Deconvolution Problems

1991/09/01 by Jianqing Fan · 15 citations
Computer Science · Engineering · Mathematics · #Applied mathematics #Blind deconvolution #Combinatorics #Computer science #Convergence (economics) #Deconvolution #Estimator #Gaussian Processes and Bayesian Inference #Kernel (algebra) #Kernel density estimation #Mathematical analysis #Mathematics #Nonparametric statistics #Random variable #Rate of convergence #Smoothness #Sparse and Compressive Sensing Techniques #Statistical Methods and Inference #Statistics

paper · pdf · doi:10.1214/aos/1176348248

openalex publication_date 1991/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03

Abstract

Deconvolution problems arise in a variety of situations in statistics. An interesting problem is to estimate the density f of a random variable X based on n i.i.d. observations from Y = X + ε, where ε is a measurement error with a known distribution. In this paper, the effect of errors in variables of nonparametric deconvolution is examined. Insights are gained by showing that the difficulty of deconvolution depends on the smoothness of error distributions: the smoother, the harder. In fact, there are two types of optimal rates of convergence according to whether the error distribution is ordinary smooth or supersmooth. It is shown that optimal rates of convergence can be achieved by deconvolution kernel density estimators.

Citations

• Optimal Global Rates of Convergence for Nonparametric Regression

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