Inference for thinned point processes, with application to Cox processes

Given i.i.d. point processes N1, N2,..., let the observations be p-thinnings N'1, N'2,..., where p is a function from the underlying space E (a compact metric space) to [0, 1], whose interpretation is that a point of Ni at x is retained with probability p(x) and deleted with probability 1-p(x). Strongly consistent estimators of the thinning function p and the Laplace functional LN(f) = E[e-N(f)] of the Ni are constructed; associated "central limit" properties are given. Tests are presented, for the case when the Ni and N'i are both observable, of the hypothesis that the N'i are p-thinnings of the Ni. State estimation techniques are developed for the case where the Ni are Cox processes directed by unobservable random measures Mi; these techniques yield minimum mean-squared error estimators, based on observation of only the thinned processes N'i of the Ni and the directing measures Mi. Limit theorems for empirical Laplace functionals of point processes are given.