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

Let X1,…,XnX1,…,Xn be i.i.d. observations, where Xi=Yi+σnZiXi=Yi+σnZi and the YY’s and ZZ’s are independent. Assume that the YY’s are unobservable and that they have the density ff and also that the ZZ’s have a known density kk. Furthermore, let σnσn depend on nn and let σn→0σn→0 as n→∞n→∞. We consider the deconvolution problem, i.e. the problem of estimation of the density ff based on the sample X1,…,XnX1,…,Xn. A popular estimator of ff in this setting is the deconvolution kernel density estimator. We derive its asymptotic normality under two different assumptions on the relation between the sequence σnσn and the sequence of bandwidths hnhn. 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→0σn→0 have to be preferred to the models with fixed σσ.