Penalized maximum likelihood estimation for a function of the intensity of a Poisson point process

Let f: [a,b] → ℝ be an unknown 2 times differentiable function and consider M to be an α- homogeneous Poisson process on Graf(f). The goal is to estimate f having a sample of the inhomogeneous Poisson process N constructed by dislocating each point of M perpendicularly to Graf(f) by a normal random variable with zero mean and constant variance σ2. The exact formulas for the mean measure and the intensity function of N are obtained. Then, the function f is estimated directly using a hybrid spline approach to penalized maximum likelihood. Simulation results indicate the procedure to be consistent as α → ∞ and σ2 → 0. © 2006 Springer Science+Business Media, LLC.