Institute of Mathematical Statistics Lecture Notes - Monograph Series

Model selection for Poisson processes

Lucien Birgé

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Our purpose in this paper is to apply the general methodology for model selection based on T-estimators developed in Birgé to the particular situation of the estimation of the unknown mean measure of a Poisson process. We introduce a Hellinger type distance between finite positive measures to serve as our loss function and we build suitable tests between balls (with respect to this distance) in the set of mean measures. As a consequence of the existence of such tests, given a suitable family of approximating models, we can build T-estimators for the mean measure based on this family of models and analyze their performances. We provide a number of applications to adaptive intensity estimation when the square root of the intensity belongs to various smoothness classes. We also give a method for aggregation of preliminary estimators.

Chapter information

Eric A. Cator, Geurt Jongbloed, Cor Kraaikamp, Hendrik P. Lopuhaä, Jon A. Wellner, eds., Asymptotics: Particles, Processes and Inverse Problems: Festschrift for Piet Groeneboom (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2007), 32-64

First available in Project Euclid: 4 December 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M30: Spatial processes 62G05: Estimation
Secondary: 62G10: Hypothesis testing 41A45: Approximation by arbitrary linear expressions 41A46: Approximation by arbitrary nonlinear expressions; widths and entropy

adaptive estimation aggregation intensity estimation model selection Poisson processes robust tests

Copyright © 2007, Institute of Mathematical Statistics


Birgé, Lucien. Model selection for Poisson processes. Asymptotics: Particles, Processes and Inverse Problems, 32--64, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2007. doi:10.1214/074921707000000265.

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