On Smooth Change-Point Location Estimation for Poisson Processes

We are interested in estimating the location of what we call "smooth change-point" from $n$ independent observations of an inhomogeneous Poisson process. The smooth change-point is a transition of the intensity function of the process from one level to another which happens smoothly, but over such a small interval, that its length $\delta_n$ is considered to be decreasing to $0$ as $n\to+\infty$. We show that if $\delta_n$ goes to zero slower than $1/n$, our model is locally asymptotically normal (with a rather unusual rate $\sqrt{\delta_n/n}$), and the maximum likelihood and Bayesian estimators are consistent, asymptotically normal and asymptotically efficient. If, on the contrary,~$\delta_n$ goes to zero faster than $1/n$, our model is non-regular and behaves like a change-point model. More precisely, in this case we show that the Bayesian estimators are consistent, converge at rate $1/n$, have non-Gaussian limit distributions and are asymptotically efficient. All these results are obtained using the likelihood ratio analysis method of Ibragimov and Khasminskii, which equally yields the convergence of polynomial moments of the considered estimators. However, in order to study the maximum likelihood estimator in the case where $\delta_n$ goes to zero faster than $1/n$, this method cannot be applied using the usual topologies of convergence in functional spaces. So, this study should go through the use of an alternative topology and will be considered in a future work.