On Smooth Change-Point Location Estimation for Poisson Processes

TL;DR

This paper investigates the estimation of smooth change-point locations in inhomogeneous Poisson processes, revealing different asymptotic behaviors depending on how quickly the transition interval shrinks as the number of observations increases.

Contribution

It characterizes the asymptotic properties of estimators for smooth change-points, distinguishing between regular and non-regular cases based on the rate at which the transition interval diminishes.

Findings

For slow decay of the transition interval, the model is locally asymptotically normal with efficient estimators.

For fast decay, the model behaves like a change-point model with non-Gaussian limit distributions.

Bayesian estimators are consistent and asymptotically efficient in both regimes.

Abstract

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.