ar X iv : 0 70 9 . 34 13 v 1 [ m at h . ST ] 2 1 Se p 20 07 Deconvolution for an atomic distribution

Let X1, . . . , Xn be i.i.d. observations, where Xi = Yi + Zi and Yi and Zi are independent. Assume that unobservable Y ’s are distributed as a random variable UV, where U has a Bernoulli distribution with probability of zero equal to p and V has a distribution function F with density f and U and V are independent. Furthermore, let the random variables Zi have the standard normal distribution. Based on a sample X1, . . . , Xn, we consider the problem of estimation of the density f and the probability p. We propose a kernel type deconvolution estimator for f and derive its asymptotic normality at a fixed point. A consistent estimator for p is given as well. Our results demonstrate that our estimator behaves very much like the kernel type deconvolution estimator in the classical deconvolution problem.