2007/05/31 by Jan Johannes · 1 citation
Computer Science · Mathematics · Medicine · #Distributed Sensor Networks and Detection Algorithms #Medical Imaging Techniques and Applications #Statistical Methods and Inference #math.ST #msc:42A38 #msc:62G05 #msc:62G07 #stat.TH
paper · pdf · doi:10.1214/08-aos652
published as Annals of Statistics 2009, Vol. 37, No. 5A, 2301-2323 · Published in at http://dx.doi.org/10.1214/08-AOS652 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2009/07/15 · arxiv created 2009/08/21 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/03
We consider the problem of estimating a density fX using a sample Y1, …, Yn from fY=fX⋆fε, where fε is an unknown density. We assume that an additional sample ε1, …, εm from fε is observed. Estimators of fX and its derivatives are constructed by using nonparametric estimators of fY and fε and by applying a spectral cut-off in the Fourier domain. We derive the rate of convergence of the estimators in case of a known and unknown error density fε, where it is assumed that fX satisfies a polynomial, logarithmic or general source condition. It is shown that the proposed estimators are asymptotically optimal in a minimax sense in the models with known or unknown error density, if the density fX belongs to a Sobolev space Hp and fε is ordinary smooth or supersmooth.