On Properties of Estimators in non Regular Situations for Poisson Processes

TL;DR

This paper reviews how the properties of maximum likelihood and Bayesian estimators for inhomogeneous Poisson processes change when standard regularity conditions are violated, highlighting non-regular estimation scenarios.

Contribution

It provides a comprehensive review of estimator behaviors in non-regular Poisson process models, extending classical results to irregular cases.

Findings

Estimator properties vary significantly without regularity conditions.

MLE and Bayesian estimators may lose consistency or efficiency.

Results are based on general theorems by Ibragimov and Khasminskii.

Abstract

We consider the problem of parameter estimation by observations of inhomogeneous Poisson process. It is well-known that if the regularity conditions are fulfilled then the maximum likelihood and Bayesian estimators are consistent, asymptotically normal and asymptotically efficient. These regularity conditions can be roughly presented as follows: a) the intensity function of observed process belongs to known parametric family of functions, b) the model is identifiable, c) the Fisher information is positive continuous function, d) the intensity function is sufficiently smooth with respect to the unknown parameter, e) this parameter is an interior point of the interval. We are interested in the properties of estimators when these regularity conditions are not fulfilled. More precisely, we preset a review of the results which correspond to the rejection of these conditions one by one and we show how the properties of the MLE and Bayesian estimators change. The proofs of these results are essentially based on some general results by Ibragimov and Khasminskii.