Asymptotic properties of estimators for the parameters of spatial inhomogeneous poisson point-processes
Consider a spatial point pattern realized from an inhomogeneous Poisson process on a bounded Borel set A subset-of R(d), with intensity function lambda (s; theta), where theta is-an-element-of theta subset-of R(k). In this article, we show that the maximum likelihood estimator theta(A) and the Bayes estimator theta(A) are consistent, asymptotically normal, and asymptotically efficient as the sample region A up R(d). These results extend asymptotic results of Kutoyants (1984), proved for an inhomogeneous Poisson process on [0, T] subset-of R, where T --> infinity. They also formalize (and extend to the multiparameter case) results announced by Krickeberg (1982), for the spatial domain R(d). Furthermore, a Cramer-Rao lower bound is found for any estimator theta(A)* of theta. The asymptotic properties of theta(A) and theta(A) are considered for modulated (Cox (1972)), and linear Poisson processes.
Rathbun, S. & Cressie, N. A. (1994). Asymptotic properties of estimators for the parameters of spatial inhomogeneous poisson point-processes. Advances In Applied Probability, 26 (1), 122-154.