## The Annals of Statistics

### On the Optimal Rates of Convergence for Nonparametric Deconvolution Problems

Jianqing Fan

#### 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.

#### Article information

Source
Ann. Statist., Volume 19, Number 3 (1991), 1257-1272.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176348248

Digital Object Identifier
doi:10.1214/aos/1176348248

Mathematical Reviews number (MathSciNet)
MR1126324

Zentralblatt MATH identifier
0729.62033

JSTOR