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

 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. Received: 24 April 2001 / Revised version: 9 October 2002 / Published online: 15 April 2003 Mathematics Subject Classification (2000): 60E15, 62G05, 62G07 Key words or phrases: Inhomogeneous Poisson process – Concentration inequalities – Model selection – Penalized projection estimator – Adaptive estimation