Deconvolution for an atomic distribution: rates of convergence

Shota Gugushvili, Bert van Es, Peter Spreij
2011 Journal of nonparametric statistics (Print)  
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
more » ... nsider 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.
doi:10.1080/10485252.2011.576763 fatcat:uzoqn5d6mfb57a47zikvbn24o4