ASYMPTOTIC PROPERTIES OF ESTIMATORS FOR THE PARAMETERS OF SPATIAL INHOMOGENEOUS POISSON POINT-PROCESSES

Consider a spatial point pattern realized from an inhomogeneous Poisso n process on a bounded Borel set A subset-of R(d), with intensity func tion lambda (s; theta), where theta is-an-element-of theta subset-of R (k). In this article, we show that the maximum likelihood estimator th eta(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 fo r 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). F urthermore, a Cramer-Rao lower bound is found for any estimator theta( A) of theta. The asymptotic properties of theta(A) and theta(A) are c onsidered for modulated (Cox (1972)), and linear Poisson processes.