Deconvolution for an atomic distribution: rates of convergence

Shota Gugushvili, Bert van Es and Peter Spreij

Journal of Nonparametric Statistics, 2011, vol. 23, issue 4, 1003-1029

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.

Date: 2011
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DOI: 10.1080/10485252.2011.576763

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