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On the optimal rates of convergence for nonparametric deconvolution problems. (English) Zbl 0729.62033
Summary: 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 \(Y=X+\epsilon\), where \(\epsilon\) is a measurement error with a known distribution. 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.

MSC:
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62C25 Compound decision problems in statistical decision theory
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