Deconvolution for an atomic distribution

van Es, Bert; Gugushvili, Shota; Spreij, Peter

Deconvolution for an atomic distribution

Authors

van Es, Bert

Gugushvili, Shota

Spreij, Peter

Description

Let $X_1,...,X_n$ 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 $X_1,..., X_n,$ 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.
Published in at http://dx.doi.org/10.1214/07-EJS121 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org)