On properties of estimators in nonregular situations for Poisson processes

Abstract

We consider the problem of parameter estimation by observations of an inhomogeneous Poisson process. It is well known that if 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 the observed process belongs to a known parametric family of functions, (b) the model is identifiable, (c) the Fisher information is a positive continuous function, (d) the intensity function is sufficiently smooth with respect to the unknown parameter, and (e) this parameter is an interior point of the interval. We are interested in properties of estimators for which these regularity conditions are not fulfilled. More precisely, we present a review of results which correspond to the rejection of these conditions one by one and show how properties of the MLE and Bayesian estimators change. The proofs of these results are essentially based on some general results by Ibragimov and Khasminskii. Bibliography: 9 titles.