A consistent nonparametric density estimator for the deconvolution problem. (English) Zbl 0694.62017

Summary: The problem of nonparametric estimation of a probability density function when the sample observations are contaminated with random noise is studied. A particular estimator \(\hat f_ n(x)\) is proposed which uses kernel-density and deconvolution techniques. The estimator \(\hat f_ n(x)\) is shown to be uniformly consistent, and its appearance and properties are affected by constants \(M_ n\) and \(h_ n\) which the user may choose. The optimal choices of \(M_ n\) and \(h_ n\) depend on the sample size n, the noise distribution, and the true distribution which is being estimated. Particular selections for \(M_ n\) and \(h_ n\) which minimize upper-bound functions of the mean squared error for \(\hat f_ n(x)\) are recommended.


62G05 Nonparametric estimation
62E20 Asymptotic distribution theory in statistics
65C05 Monte Carlo methods
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[1] Blum, Estimation of a mixing distribution function, Ann. Probab. 5 pp 200– (1977) · Zbl 0358.60040
[2] Blum, A Fourier inversion method of estimation of a density and its derivative, J. Austral. Math. Soc. Ser. A 23 pp 166– (1977) · Zbl 0357.62025
[3] Choi, An estimation procedure for mixtures of distributions, J. Roy. Statist. Soc. Ser. B 30 pp 444– (1968) · Zbl 0187.15804
[4] Crump, A new algorithm for inversion of aerosol size distribution data, Aerosol Sci. and Technol 1 pp 15– (1982)
[5] Deely, Construction of sequences estimating the mixing distribution, Ann. Math. Statist. 39 pp 286– (1968) · Zbl 0174.22302
[6] Devroye, A note on consistent deconvolution in density estimation, Canad. J. Statist. 17 (1989)
[7] Liu, M. C., and Taylor, R. L. (1987). Simulations and computations of nonparametric density estimates for the deconvulution problem. Statistics Technical Report 74, University of Georgia; J. Statist. Comput. Simulation, to appear.
[8] Mendelsohn, Deconvolution of microfluorometric histograms with B-splines, J. Amer. Statist. Assoc. 77 pp 748– (1982)
[9] O’Bryan, An empirical Bayes estimation problem with nonidentical components involving normal distributions, Comm. Statist. 4 (11) pp 1033– (1975)
[10] O’Bryan, Rates in the empirical Bayes estimation problem with non-identical components - case of normal distributions, Ann. Inst. Statist. Math. 28A pp 389– (1976)
[11] Parzen, Estimation of a probability density function and mode, Ann. Math. Statist. 33 pp 1065– (1962) · Zbl 0116.11302
[12] Rice, Smoothing splines: Regression, derivatives and deconvolution, Ann. Statist. 11 pp 141– (1983) · Zbl 0535.41019
[13] Rice, Choice of smoothing parameter in deconvolution problems, Contemp. Math. 59 (1986) · Zbl 0623.62032
[14] Stefanski, L., and Carroll, R. J. (1989). Deconvolution-kernel density estimators, Statistics, to appear. · Zbl 0697.62035
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