2004/09/24 by Cristina Butucea, Butucea, Cristina, Alexandre B. Tsybakov +1 · 1 citation
Computer Science · Mathematics · #62G05 #62G20 #Bayesian Methods and Mixture Models #FOS: Mathematics #Statistical Methods and Bayesian Inference #Statistical Methods and Inference #Statistics Theory (math.ST) #math.ST #msc:62G05 #msc:62G20 #stat.TH
paper · pdf · doi:10.48550/arxiv.math/0409471
arxiv created 2004/09/24 · openalex publication_date 2004/09/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider estimation of the common probability density f of i.i.d. random variables Xi that are observed with an additive i.i.d. 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 is supposed to be 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 that is optimal in sharp asymptotical minimax sense on A simultaneously under the pointwise and the \mathbbL2-risks. The variance of the estimators turns out to be asymptotically negligible w.r.t. its squared bias. For r<s/2 we construct a sharp adaptive estimator of f. We discuss some effects of dominating bias, such as superefficiency of minimax estimators.