Deconvolution for an atomic distribution

Abstract

Let X₁,…,X_n be i.i.d. observations, where X_i=Y_i+σZ_i and Y_i and Z_i are independent. Assume that unobservable Y’s are distributed as a random variable UV, where U and V are independent, U has a Bernoulli distribution with probability of zero equal to p and V has a distribution function F with density f. Furthermore, let the random variables Z_i have the standard normal distribution and let σ>0. Based on a sample X₁,…,X_n, 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.