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Asymptotic Properties of Estimators for the Parameters of Spatial Inhomogeneous Poisson Point Processes
- Part of
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- Journal:
- Advances in Applied Probability / Volume 26 / Issue 1 / March 1994
- Published online by Cambridge University Press:
- 01 July 2016, pp. 122-154
- Print publication:
- March 1994
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- Article
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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.
, with intensity function 
. 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, 
, where 
. Furthermore, a Cramér–Rao lower bound is found for any estimator 
of 
and 
are considered for modulated (Cox (1972)), and linear Poisson processes.