Geodesic
Sharp optimality in density deconvolution with dominating bias. II
Teoriâ veroâtnostej i ee primeneniâ, Tome 52 (2007) no. 2, pp. 336-349 Cet article a éte moissonné depuis la source Math-Net.Ru

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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, 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 we construct a sharp adaptive estimator of $f$.
Keywords: nonparametric density estimation, infinitely differentiable functions, exact constants in nonparametric smoothing, minimax risk, adaptive curve estimation.
Mots-clés : deconvolution 
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     author = {C. Butucea and A. Tsybakov},
     title = {Sharp optimality in density deconvolution with dominating {bias.~II}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {336--349},
     year = {2007},
     volume = {52},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_2007_52_2_a4/}
}
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C. Butucea; A. Tsybakov. Sharp optimality in density deconvolution with dominating bias. II. Teoriâ veroâtnostej i ee primeneniâ, Tome 52 (2007) no. 2, pp. 336-349. http://geodesic.mathdoc.fr/item/TVP_2007_52_2_a4/

[1] Brown L. D., Low M. G., Zhao L. H., “Superefficiency in nonparametric function estimation”, Ann. Statist., 25:6 (1997), 2607–2625 | DOI | MR | Zbl

[2] Butucea C., Tsybakov A. B., “Sharp optimality for density deconvolution with dominating bias. I”, Teoriya veroyatn. i ee primen., 52:1 (2007), 111–128 | MR

[3] Tsybakov A. B., Introduction à l'estimation non-paramétrique, Springer-Verlag, Berlin, 2004, 175 pp. | MR