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(-alpha|u|(r)) as vertical bar u vertical bar -> infinity, where alpha > 0, r > 0. The noise density assumed known and such that its characteristic function decays as exp(-beta vertical bar u vertical bar(s)), as vertical bar u vertical bar -> infinity, 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 L-2 risks. For r < s/ 2 we construct a sharp adaptive estimator of f.