Deconvolution for an atomic distribution: rates of convergence
Abstract
Let X1,..., Xn be i.i.d.\ copies of a random variable X=Y+Z, where Xi=Yi+Zi, and Yi and Zi are independent and have the same distribution as Y and Z, respectively. Assume that the random variables Yi'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 X1,...,Xn, 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.