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On optimal estimation of the mode in nonparametric deconvolution problems

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  • Barbara Wieczorek

Abstract

This work deals with the problem of estimating the mode in nonparametric deconvolution models. First, given n i.i.d. observations from Y=X+ϵ, we consider estimating the mode θ of a density function of some random variable X. Second, we consider the errors-in-variables regression model, where we are interested in the mode of m(x)=E(Z|X=x), where n i.i.d. observations from (Y, Z) with Y=X+ϵ are given. In both cases, we assume the distribution of ϵ to be ordinary smooth. The mode estimator ˆθn is defined via maximising over a curve estimator of the kernel type. In both deconvolution models, we obtain rates for the quadratic risk of ˆθn, depending on the smoothness of the underlying curve and the degree of ill-posedness of the deconvolution problem. Further, we show that these rates are optimal, considering one-dimensional subproblems in the class of functions studied.

Suggested Citation

  • Barbara Wieczorek, 2010. "On optimal estimation of the mode in nonparametric deconvolution problems," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(1), pages 65-80.
    • Handle: RePEc:taf:gnstxx:v:22:y:2010:i:1:p:65-80
    DOI: 10.1080/10485250903121626
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    References listed on IDEAS

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    1. Stefanski, Leonard A., 1990. "Rates of convergence of some estimators in a class of deconvolution problems," Statistics & Probability Letters, Elsevier, vol. 9(3), pages 229-235, March.
    2. Vieu, Philippe, 1996. "A note on density mode estimation," Statistics & Probability Letters, Elsevier, vol. 26(4), pages 297-307, March.
    3. Rachdi, Mustapha & Sabre, Rachid, 2000. "Consistent estimates of the mode of the probability density function in nonparametric deconvolution problems," Statistics & Probability Letters, Elsevier, vol. 47(2), pages 105-114, April.
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