Poisson inverse problems

• 成果类型: Article
• 署名作者: Antoniadis, Anestis; Bigot, Jeremie
• 署名单位: Communaute Universite Grenoble Alpes; Universite Grenoble Alpes (UGA); Universite de Toulouse; Universite Toulouse III - Paul Sabatier
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053606000000687
• 发表日期: 2006
• 页码: 2132-2158
• 关键词: size distributions; emission tomography; DENSITY-ESTIMATION; wavelet; intensity; models; approximation; DECOMPOSITION; CONVERGENCE

摘要

In this paper we focus on nonparametric estimators in inverse problems for Poisson processes involving the use of wavelet decompositions. Adopting an adaptive wavelet Galerkin discretization, we find that our method combines the well-known theoretical advantages of wavelet-vaguelette decompositions for inverse problems in terms of optimally adapting to the unknown smoothness of the solution, together with the remarkably simple closed-form expressions of Galerkin inversion methods. Adapting the results of Barron and Sheu [Ann. Statist. 19 (1991) 1347-1369] to the context of log-intensity functions approximated by wavelet series with the use of the Kullback-Leibler distance between two point processes, we also present an asymptotic analysis of convergence rates that justifies our approach. In order to shed some light on the theoretical results obtained and to examine the accuracy of our estimates in finite samples, we illustrate our method by the analysis of some simulated examples.