Estimation and statistical inference for space-time point processes

A spatial point pattern consists of the locations of events in some sample window A ⊂ IR[superscript] d. Events may be trees in a forest, towns in a country, or epicenters of earthquakes. Assume that the locations of events are realized from some spatial point process, a random mechanism for locating events in space. One goal of spatial statistics is to model the underlying stochastic process and thus reduce a complex point pattern to one or more parameter estimates that may have some scientific interpretation;A number of techniques has been proposed for estimating the parameters of spatial point processes. However, very little is known about the statistical properties of these estimators. This research considers parameter estimation for an inhomogeneous Poisson process on A ⊂ IR[superscript] d with intensity function [lambda](s;[theta]). It is shown that the maximum likelihood estimator [theta][subscript] A and the Bayes estimator ~[theta][subscript] A are consistent, asymptotically normal, and asymptotically efficient, as the sample region A↑IR[superscript] d. This extends asymptotic results of Kutoyants (1984) for an inhomogeneous Poisson process on (0, T) ⊂IR, where T → [infinity]. Furthermore, a Cramer-Rao lower bound is found for any estimator of [theta]. The asymptotic properties of [theta][subscript] A and ~[theta][subscript] A are considered for a modulated Poisson process (Cox, 1972) and a linear Poisson process. An example is given to show that there exists an inhomogeneous Poisson process that has no consistent estimators;The marked spatial point pattern of trees and their diameters is the result of a dynamic process that takes place over time as well as space. Such marked point patterns are realizations of marked space-time survival point processes, where trees are born at some random location and time, and then live and grow for a random length of time. A model for a marked space-time survival point process is fit to data from a longleaf pine forest in southern Georgia. The space-time survival point process is divided into three components, a birth process, a growth process, and survival process, and each of the component processes is analyzed separately. By using this reductionist approach, questions concerning each individual process can be addressed that might not have been answerable had all processes been combined in the model;References. (1) Cox, D. R. (1972). The statistical analysis of dependencies in point processes. In P.A.W. Lewis (ed.), Stochastic Point Processes (pp. 55-66). New York: Wiley. (2) Kutoyants, Yu. A. (1984). Parameter Estimation for Stochastic Processes. Berlin: Heldermann Verlag