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Teoriya Veroyatnostei i ee Primeneniya, 2007, Volume 52, Issue 2, Pages 336–349
DOI: https://doi.org/10.4213/tvp175
(Mi tvp175)
 

This article is cited in 54 scientific papers (total in 54 papers)

Sharp optimality in density deconvolution with dominating bias. II

C. Butuceaa, A. Tsybakovb

a Université Paris X
b Université Pierre & Marie Curie, Paris VI
References:
Abstract: We consider estimation of the common probability density $f$ of iid random variables $X_i$ that are observed with an additive iid noise. We assume that the unknown density $f$ belongs to a class $\mathcal{A}$ of densities whose characteristic function is described by the exponent $\exp(-\alpha |u|^r)$ as $|u|\to\infty$, where $\alpha>0$, $r>0$. The noise density is assumed known and such that its characteristic function decays as $\exp(-\beta|u|^s)$, as $|u|\to\infty$, where $\beta>0$, $s>0$. Assuming that $r<s$, we suggest a kernel-type estimator, whose variance turns out to be asymptotically negligible with respect to its squared bias under both pointwise and $\mathbb{L}_2$ risks. For $r<s/2$ we construct a sharp adaptive estimator of $f$.
Keywords: deconvolution, nonparametric density estimation, infinitely differentiable functions, exact constants in nonparametric smoothing, minimax risk, adaptive curve estimation.
Received: 30.08.2004
Revised: 27.06.2005
English version:
Theory of Probability and its Applications, 2008, Volume 52, Issue 2, Pages 237–249
DOI: https://doi.org/10.1137/S0040585X97982992
Bibliographic databases:
Language: English
Citation: C. Butucea, A. Tsybakov, “Sharp optimality in density deconvolution with dominating bias. II”, Teor. Veroyatnost. i Primenen., 52:2 (2007), 336–349; Theory Probab. Appl., 52:2 (2008), 237–249
Citation in format AMSBIB
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  • https://www.mathnet.ru/eng/tvp/v52/i2/p336
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    This publication is cited in the following 54 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теория вероятностей и ее применения Theory of Probability and its Applications
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