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Asymptotic Properties of Estimators for the Parameters of Spatial Inhomogeneous Poisson Point Processes

Part of: Distribution theory Inference from stochastic processes Stochastic processes Parametric inference

Published online by Cambridge University Press:  01 July 2016

Stephen L. Rathbun  and
Noel Cressie
Stephen L. Rathbun*
Affiliation:
University of Georgia
Noel Cressie*
Affiliation:
Iowa State University
*
* Postal address: Department of Statistics, University of Georgia, Athens, GA 30602–1952, USA.
** Postal address: Statistical Laboratory and Department of Statistics, Snedecor Hall, Ames, IA 50011–1210, USA.
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Abstract

Consider a spatial point pattern realized from an inhomogeneous Poisson process on a bounded Borel set , with intensity function λ (s; θ), where . In this article, we show that the maximum likelihood estimator and the Bayes estimator are consistent, asymptotically normal, and asymptotically efficient as the sample region . These results extend asymptotic results of Kutoyants (1984), proved for an inhomogeneous Poisson process on [0, T] , where T →∞. They also formalize (and extend to the multiparameter case) results announced by Krickeberg (1982), for the spatial domain . Furthermore, a Cramér–Rao lower bound is found for any estimator of θ. The asymptotic properties of and are considered for modulated (Cox (1972)), and linear Poisson processes.

Keywords

SPATIAL POINT PROCESSMAXIMUM LIKELIHOOD ESTIMATORBAYES ESTIMATORCONSISTENCYASYMPTOTIC NORMALITYASYMPTOTIC EFFICIENCYMODULATED POISSON PROCESSCRAMÉR–RAO INEQUALITY

MSC classification

Primary: 62M09: Non-Markovian processes
Secondary: 60G55: Point processes 62E20: Asymptotic distribution theory 62F12: Asymptotic properties of estimators
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 26 , Issue 1 , March 1994 , pp. 122 - 154
Copyright
Copyright © Applied Probability Trust 1994 

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