Deconvolution for an atomic distribution: Rates of convergence

S. Gugushvili, Bert Es, van, P.J.C. Spreij

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Abstract

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. Keywords: atomic distribution, deconvolution, Fourier inversion, kernel smoothing, mean square error, mean integrated square error, optimal convergence rate
Original languageEnglish
Pages (from-to)1003-1029
JournalJournal of Nonparametric Statistics
Volume23
Issue number4
DOIs
Publication statusPublished - 2011

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