Asymptotic normality of the deconvolution kernel density estimator under the vanishing error variance

Let X-1,..., X-n be i.i.d. observations, where X-i = Y-i + sigma(n)Z(i) and the Y's and Z's are independent. Assume that the Y's are unobservable and that they have the density f and also that the Z's have a known density k. Furthermore, let sigma(n) depend on n and let sigma(n) -> 0 as n -> infinity. We consider the deconvolution problem, i.e. the problem of estimation of the density f based on the sample X-1,, X-n. A popular estimator of f in this setting is the deconvolution kernel density estimator. We derive its asymptotic normality under two different assumptions on the relation between the sequence sigma(n) and the sequence of bandwidths h(n). We also consider several simulation examples which illustrate different types of asymptotics corresponding to the derived theoretical results and which show that there exist situations where models with sigma(n) -> 0 have to be preferred to the models with fixed sigma. (C) 2009 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved.