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Estimation adaptative de l'intensité de certains processus ponctuels par sélection de modèle.

Abstract : In this thesis, we want to adapt model selection technics to
the problem of estimating the intensity of point processes. More precisely, we
want to prove that penalized projection estimators of the intensity are
adaptive either in a family of projection estimators or for the minimax risk.
We restrict ourselves to the inhomogeneous Poisson processes and to the
counting processes with Aalen's multiplicative intensity.In both cases, we want to prove an oracle type inequality which
shows that penalized projection estimators do quite as well as the best
projection estimator for a given family of models. The key to derive oracle
type inequalities is the concentration phenomena and more precisely the
knowledge of some exponential inequalities which allow us to control in
probability the deviations of chi-square type statistics upon their mean. We
prove two different concentration inequalities. The first one is only valid
for Poisson processes. It has the same orders of magnitude as the one due to
M. Talagrand for suprema of empirical processes. The second one does not
achieve these orders of magnitude but is valid for more general counting
processes.This last result uses martingales technics which can be applied also
to derive concentration inequalities for degenerate U-statistics of order 2
also for double integrals with respect to a centred Poisson measure. We also compute some lower bounds for minimax risks and we show
that penalized projection estimators reach these lower bounds up to some
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Contributor : Patricia Reynaud-Bouret Connect in order to contact the contributor
Submitted on : Friday, June 23, 2006 - 10:13:01 AM
Last modification on : Thursday, March 17, 2022 - 10:08:16 AM
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  • HAL Id : tel-00081412, version 1


Patricia Reynaud-Bouret. Estimation adaptative de l'intensité de certains processus ponctuels par sélection de modèle.. Mathématiques [math]. Université Paris Sud, Orsay, 2002. Français. ⟨tel-00081412⟩



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