  Let $X_1,...,X_n$ be i.i.d. observations, where $X_i=Y_i+\sigma Z_i$ 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 $\sigma>0.$ Based
on a sample $X_1,..., X_n,$ 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.
