Asymptotic properties of the maximum likelihood estimator for spatio-temporal point processes

Abstract

Consider a spatio-temporal point process whose events occur at times in the interval [0, T] and at corresponding locations in a region X. Such processes can be modeled through their conditional intensity function Λ( s ~ , t; θ ~ ); 0 ⩽ t ⩽ T, s ~ ∈ X . This article shows that the Maximum likelihood estimator θ ̂ T is consistent and asymptotically normally distributed as T → ∞. These results extend those of Ogata ( Ann. Inst. Statist. Math. 30A (1978), 243–261), who considered purely temporal pointproceses. The asymptotic properties of θ ̂ T are considered for a spatio-temporal self-exiciting point process. Methods for modeling spatio-temporal point patterns are illustrated on seismological data.