Estimation of Cusp Location by Poisson Observations

We consider an inhomogeneous Poisson process X on [0, T]. The intensity function of X is supposed to be regular on [0, T] except at the point θ, in which it has a singularity (a cusp) of order p. We suppose that we know the shape of the intensity function, but not the location (given by the parameter θ) of the point of cusp. We consider the problem of estimation of this location (shift) parameter θ based on n observations of the process X. We study the maximum likelihood estimator and the Bayesian estimators. We show that these estimators are consistent, their rate of convergence is n1/(2p+1), they have different limit distributions, and the Bayesian estimators are asymptotically efficient.