Sharp Optimality in Density Deconvolution with Dominating Bias. I

We consider estimation of the common probability density f of independent identically distributed random variables Xᵢ that are observed with an additive independent identically distributed noise. We assume that the unknown density f belongs to a class A of densities whose characteristic function is described by the exponent exp(-α |u|ʳ) as |u|→∞, where α>0, $r>0$. The noise density assumed known and such that its characteristic function decays as exp (-β|u|ˢ), as |u|→∞, where β>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 L₂ risks. For $r < s/2$ we construct a sharp adaptive estimator of f.