Estimation of the location of a 0-type or ∞-type singularity by Poisson observations

摘要

We consider an inhomogeneous Poisson process X on [0, T]. The intensity function of X is supposed to be strictly positive and smooth on [0, T] except at the point θ, in which it has either a 0-type singularity (tends to 0 like |x| p , p(0, 1)), or an ∞-type singularity (tends to ∞ like |x| p , p(−1, 0)). We suppose that we know the shape of the intensity function, but not the location of the singularity. We consider the problem of estimation of this location (shift) parameter θ based on n observations of the process X. We study the Bayesian estimators and, in the case p>0, the maximum-likelihood estimator. We show that these estimators are consistent, their rate of convergence is n 1/(p+1), they have different limit distributions, and the Bayesian estimators are asymptotically efficient.