Inhomogeneous Poisson process rate function inference from dead-time limited observations

2017 ◽  
Vol 34 (5) ◽  
pp. 770 ◽  
Author(s):  
Gunjan Verma ◽  
Robert J. Drost
Keyword(s):  
Poisson Process ◽  
Dead Time ◽  
Rate Function ◽  
Process Rate ◽  
Inhomogeneous Poisson Process

Misspecified change-point estimation problem for a Poisson process

2001 ◽  
Vol 38 (A) ◽  
pp. 122-130 ◽  
Author(s):  
Ali S. Dabye ◽  
Yury A. Kutoyants
Keyword(s):  
Maximum Likelihood ◽  
Poisson Process ◽  
Change Point ◽  
Maximum Likelihood Estimate ◽  
Intensity Function ◽  
Point Estimation ◽  
Lower Function ◽  
Inhomogeneous Poisson Process ◽  
Change Point Estimation ◽  
Unknown Point

Consider an inhomogeneous Poisson process X on [0, T] whose unknown intensity function ‘switches' from a lower function g∗ to an upper function h∗ at some unknown point θ∗. What is known are continuous bounding functions g and h such that g∗(t) ≤ g(t) ≤ h(t) ≤ h∗(t) for 0 ≤ t ≤ T. It is shown that on the basis of n observations of the process X the maximum likelihood estimate of θ∗ is consistent for n →∞, and also that converges in law and in pth moment to limits described in terms of the unknown functions g∗ and h∗.

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

1994 ◽  
Vol 26 (1) ◽  
pp. 122-154 ◽  
Author(s):  
Stephen L. Rathbun ◽  
Noel Cressie
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Asymptotic Properties ◽  
Point Pattern ◽  
Spatial Point Pattern ◽  
Asymptotic Results ◽  
Borel Set ◽  
Inhomogeneous Poisson Process ◽  
Asymptotically Normal ◽  
Asymptotically Efficient

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.

Non-homogeneously paced random records and associated extremal processes

1978 ◽  
Vol 15 (3) ◽  
pp. 552-559 ◽  
Author(s):  
Donald P. Gaver ◽  
Patricia A. Jacobs
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Extreme Value Distribution ◽  
Rate Function ◽  
Extreme Value ◽  
Value Distribution ◽  
Homogeneous Poisson Process ◽  
Extremal Process ◽  
Extremal Processes ◽  
Non Homogeneous Poisson Process

A study is made of the extremal process generated by i.i.d. random variables appearing at the events of a non-homogeneous Poisson process, 𝒫. If 𝒫 has an exponentially increasing rate function, then records eventually occur in a homogeneous Poisson process. The size of the latest record has a classical extreme value distribution.

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

1994 ◽  
Vol 26 (01) ◽  
pp. 122-154 ◽  
Author(s):  
Stephen L. Rathbun ◽  
Noel Cressie
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Asymptotic Properties ◽  
Point Pattern ◽  
Spatial Point Pattern ◽  
Asymptotic Results ◽  
Borel Set ◽  
Inhomogeneous Poisson Process ◽  
Asymptotically Normal ◽  
Asymptotically Efficient

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.

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