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Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities. (English) Zbl 1019.62079
Summary: 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.

MSC:
62M05 Markov processes: estimation; hidden Markov models
62E15 Exact distribution theory in statistics
62G07 Density estimation
62G05 Nonparametric estimation
62M09 Non-Markovian processes: estimation
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