Sharp optimality in density deconvolution with dominating bias. I

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 $\mathcal A$ 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 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 both under the pointwise and $\mathbf L_2$ risks. For $r<s/2$ we construct a sharp adaptive estimator of $f$.