On the Optimal Rates of Convergence for Nonparametric Deconvolution Problems

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.