Penalized Maximum Likelihood Estimation For a Function of the Intensity of a Poisson Point Process

Let $f\colon [a,b]\to R$ be a smooth function and consider $M$ to be an $\alpha$-homogeneous Poisson process on $Graf(f)$. The goal is to estimate $f$ having a sample of an 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. The function $f$ is estimated directly using a hybrid spline approach to penalized maximum likelihood.