Deconvolution for an atomic distribution: rates of convergence

作者：Gugushvili Shota^*; van Es Bert; Spreij Peter

来源：Journal of Nonparametric Statistics, 2011, 23(4): 1003-1029.

DOI：10.1080/10485252.2011.576763

• atomic distribution
• deconvolution
• Fourier inversion
• kernel smoothing
• mean square error
• mean integrated square error
• optimal convergence rate

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.

• 出版日期2011