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Deconvolution for an atomic distribution: rates of convergence

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  • Shota Gugushvili
  • Bert van Es
  • Peter Spreij

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.

Suggested Citation

  • Shota Gugushvili & Bert van Es & Peter Spreij, 2011. "Deconvolution for an atomic distribution: rates of convergence," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(4), pages 1003-1029.
    • Handle: RePEc:taf:gnstxx:v:23:y:2011:i:4:p:1003-1029
    DOI: 10.1080/10485252.2011.576763
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    References listed on IDEAS

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    1. Mihee Lee & Haipeng Shen & Christina Burch & J. Marron, 2010. "Direct deconvolution density estimation of a mixture distribution motivated by mutation effects distribution," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(1), pages 1-22.
    2. Ingrid K. Glad & Nils Lid Hjort & Nikolai G. Ushakov, 2003. "Correction of Density Estimators that are not Densities," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(2), pages 415-427, June.
    3. Chen, Song X. & Delaigle, Aurore & Hall, Peter, 2010. "Nonparametric estimation for a class of Lévy processes," Journal of Econometrics, Elsevier, vol. 157(2), pages 257-271, August.
    4. Delaigle, Aurore & Hall, Peter, 2006. "On optimal kernel choice for deconvolution," Statistics & Probability Letters, Elsevier, vol. 76(15), pages 1594-1602, September.
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    1. Costa, Manon & Gadat, Sébastien & Gonnord, Pauline & Risser, Laurent, 2018. "Cytometry inference through adaptive atomic deconvolution," TSE Working Papers 18-905, Toulouse School of Economics (TSE).

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