Deconvolution for an atomic distribution : rates of convergence

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 = UV, where U and V are independent, U 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 and show that the estimators are rate-optimal.