Let X1,…,Xn be i.i.d. observations, where Xi=Yi+σZi and Yi and Zi 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 Zi have the standard normal distribution and let σ>0. 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.
"Deconvolution for an atomic distribution." Electron. J. Statist. 2 265 - 297, 2008. https://doi.org/10.1214/07-EJS121