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Poisson inverse problems. (English) Zbl 1106.62035
Summary: 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 A. R. Barron and C.-H. Sheu [ibid. 19, No. 3, 1347–1369 (1991; Zbl 0739.62027)] 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.

MSC:
62G07 Density estimation
62M05 Markov processes: estimation; hidden Markov models
65T60 Numerical methods for wavelets
65J10 Numerical solutions to equations with linear operators
62M09 Non-Markovian processes: estimation
65C60 Computational problems in statistics (MSC2010)
46N30 Applications of functional analysis in probability theory and statistics
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