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October 2006 Poisson inverse problems
Anestis Antoniadis, Jéremie Bigot
Ann. Statist. 34(5): 2132-2158 (October 2006). DOI: 10.1214/009053606000000687


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.


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Anestis Antoniadis. Jéremie Bigot. "Poisson inverse problems." Ann. Statist. 34 (5) 2132 - 2158, October 2006.


Published: October 2006
First available in Project Euclid: 23 January 2007

zbMATH: 1106.62035
MathSciNet: MR2291495
Digital Object Identifier: 10.1214/009053606000000687

Primary: 62G07
Secondary: 65J10

Keywords: adaptive estimation , Besov spaces , Galerkin inversion , integral equation , intensity function , Poisson process , wavelet thresholding , Wavelets

Rights: Copyright © 2006 Institute of Mathematical Statistics


Vol.34 • No. 5 • October 2006
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