On the Optimal Rates of Convergence for Nonparametric Deconvolution Problems

TLDR

Deconvolution problems arise in many statistical settings, such as estimating a density f of X from observations Y = X + ε when the error distribution is known. This paper investigates how measurement‑error characteristics affect nonparametric deconvolution. The authors show that deconvolution difficulty hinges on error smoothness—ordinary‑smooth errors yield one optimal rate, supersmooth errors another—and that kernel density estimators attain these optimal rates.

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 $Y = X + \varepsilon$, where $\varepsilon$ 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.