  Let $X_1,..., X_n$ be i.i.d.\ copies of a random variable $X=Y+Z,$ where $
X_i=Y_i+Z_i,$ and $Y_i$ and $Z_i$ are independent and have the same
distribution as $Y$ and $Z,$ respectively. Assume that the random variables
$Y_i$'s are unobservable and that $Y=AV,$ where $A$ and $V$ are independent,
$A$ has a Bernoulli distribution with probability of success equal to $1-p$ and
$V$ has a distribution function $F$ with density $f.$ Let the random variable
$Z$ have a known distribution with density $k.$ Based on a sample
$X_1,...,X_n,$ we consider the problem of nonparametric estimation of the
density $f$ and the probability $p.$ Our estimators of $f$ and $p$ are
constructed via Fourier inversion and kernel smoothing. We derive their
convergence rates over suitable functional classes. By establishing in a number
of cases the lower bounds for estimation of $f$ and $p$ we show that our
estimators are rate-optimal in these cases.
