Sharp Optimality in Density Deconvolution with Dominating Bias. II

• Published in: Theory of probability and its applications 2008-01, Vol.52 (2), p.237-249
• Main Authors: Butucea, C., Tsybakov, A. B.
• Format: Article
• Language: English
• Subjects: Adaptation; Bandwidths; Bias; Mathematics; Noise; Probability; Random variables
• ISSN: 0040-585X; 1095-7219; 1095-7219
• DOI: 10.1137/S0040585X97982992

We consider estimation of the common probability density $f$ of independent identically distributed random variables $X_i$ that are observed with an additive independent identically distributed noise. We assume that the unknown density $f$ belongs to a class ${\cal A}$ 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