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Probability Theory and Related Fields

, Volume 126, Issue 1, pp 103–153 | Cite as

Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities

  • Patricia Reynaud-Bouret

Abstract.

 In this paper, we establish oracle inequalities for penalized projection estimators of the intensity of an inhomogeneous Poisson process. We study consequently the adaptive properties of penalized projection estimators. At first we provide lower bounds for the minimax risk over various sets of smoothness for the intensity and then we prove that our estimators achieve these lower bounds up to some constants. The crucial tools to obtain the oracle inequalities are new concentration inequalities for suprema of integral functionals of Poisson processes which are analogous to Talagrand's inequalities for empirical processes.

Keywords

Lower Bound Poisson Process Integral Functional Empirical Process Adaptive Estimation 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Patricia Reynaud-Bouret
    • 1
  1. 1.Georgia Institute of Technology, School of Mathematics, Atlanta, GA 30332, USA. e-mail: Patricia.Reynaud@dma.ens.frUS

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