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 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 L ₂ risks. For r < s/2 we construct a sharp adaptive estimator of f.