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

Let X1,…,Xn be i.i.d. observations, where Xi=Yi+σnZi 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 σn depend on n and let σn→0 as n→∞. We consider the deconvolution problem, i.e. the problem of estimation of the density f based on the sample X1,…,Xn. 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 σn and the sequence of bandwidths hn. 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 σn→0 have to be preferred to the models with fixed σ.