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Teor. Veroyatnost. i Primenen., 2007, Volume 52, Issue 2, Pages 336–349 (Mi tvp175)  

This article is cited in 29 scientific papers (total in 29 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

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.


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English version:
Theory of Probability and its Applications, 2008, 52:2, 237–249

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Received: 30.08.2004
Revised: 27.06.2005

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
\by C.~Butucea, A.~Tsybakov
\paper Sharp optimality in density deconvolution with dominating bias.~II
\jour Teor. Veroyatnost. i Primenen.
\yr 2007
\vol 52
\issue 2
\pages 336--349
\jour Theory Probab. Appl.
\yr 2008
\vol 52
\issue 2
\pages 237--249

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    3. van Es B., Gugushvili Sh., Spreij P., “Deconvolution for an atomic distribution”, Electron. J. Stat., 2 (2008), 265–297  crossref  mathscinet  zmath  isi  scopus
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    15. Ray K., “Bayesian Inverse Problems with Non-Conjugate Priors”, Electron. J. Stat., 7 (2013), 2516–2549  crossref  mathscinet  zmath  isi  scopus
    16. Johannes J., Schwarz M., “Adaptive Circular Deconvolution by Model Selection Under Unknown Error Distribution”, Bernoulli, 19:5A (2013), 1576–1611  crossref  mathscinet  zmath  isi  scopus
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    18. Dion Ch., “New Adaptive Strategies For Nonparametric Estimation in Linear Mixed Models”, J. Stat. Plan. Infer., 150 (2014), 30–48  crossref  mathscinet  zmath  isi  scopus
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    21. Scricciolo C., “Adaptive Bayesian Density Estimation in l-P-Metrics With Pitman-Yor Or Normalized Inverse-Gaussian Process Kernel Mixtures”, Bayesian Anal., 9:2 (2014), 475–520  crossref  mathscinet  zmath  isi  elib  scopus
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    23. Chesneau C., Comte F., Mabon G., Navarro F., “Estimation of Convolution in the Model With Noise”, J. Nonparametr. Stat., 27:3 (2015), 286–315  crossref  mathscinet  zmath  isi  scopus
    24. Dedecker J., Fischer A., Michel B., “Improved Rates For Wasserstein Deconvolution With Ordinary Smooth Error in Dimension One”, Electron. J. Stat., 9:1 (2015), 234–265  crossref  mathscinet  zmath  isi  scopus
    25. Dion Ch., “Nonparametric estimation in a mixed-effect Ornstein?Uhlenbeck model”, Metrika, 79:8 (2016), 919–951  crossref  mathscinet  zmath  isi  scopus
    26. Mabon G., “Adaptive deconvolution of linear functionals on the nonnegative real line”, J. Stat. Plan. Infer., 178 (2016), 1–23  crossref  mathscinet  zmath  isi  scopus
    27. Lepski O.V., Willer T., “Lower bounds in the convolution structure density model”, Bernoulli, 23:2 (2017), 884–926  crossref  mathscinet  zmath  isi  scopus
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  • Теория вероятностей и ее применения Theory of Probability and its Applications
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