Estimation of the intensity function of an inhomogeneous Poisson process with a change‐point

2019 ◽  
Vol 47 (4) ◽  
pp. 604-618
Author(s):  
Tin Lok J. Ng ◽  
Thomas B. Murphy
Keyword(s):  
Poisson Process ◽  
Change Point ◽  
Intensity Function ◽  
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∗.

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 ∗.

On an optimal selection problem of Cowan and Zabczyk

1987 ◽  
Vol 24 (4) ◽  
pp. 918-928 ◽  
Author(s):  
F. Thomas Bruss
Keyword(s):  
Poisson Process ◽  
Continuous Time ◽  
Intensity Function ◽  
Poisson Processes ◽  
Secretary Problem ◽  
Selection Problem ◽  
Recall Condition ◽  
Optimal Selection ◽  
Multiplicative Constant ◽  
Inhomogeneous Poisson Process

Cowan and Zabczyk (1978) have studied a continuous-time generalization of the so-called secretary problem, where options arise according to a homogeneous Poisson processes of known intensity λ. They gave the complete strategy maximizing the probability of accepting the best option under the usual no-recall condition. In this paper, the solution is extended to the case where the intensity λ is unknown, and also to the case of an inhomogeneous Poisson process with intensity function λ (t), which is either supposed to be known or known up to a multiplicative constant.

On an optimal selection problem of Cowan and Zabczyk

1987 ◽  
Vol 24 (04) ◽  
pp. 918-928 ◽  
Author(s):  
F. Thomas Bruss
Keyword(s):  
Poisson Process ◽  
Continuous Time ◽  
Intensity Function ◽  
Poisson Processes ◽  
Secretary Problem ◽  
Selection Problem ◽  
Recall Condition ◽  
Optimal Selection ◽  
Multiplicative Constant ◽  
Inhomogeneous Poisson Process

Cowan and Zabczyk (1978) have studied a continuous-time generalization of the so-called secretary problem, where options arise according to a homogeneous Poisson processes of known intensity λ. They gave the complete strategy maximizing the probability of accepting the best option under the usual no-recall condition. In this paper, the solution is extended to the case where the intensity λ is unknown, and also to the case of an inhomogeneous Poisson process with intensity function λ (t), which is either supposed to be known or known up to a multiplicative constant.

Weak limit results for the extremes of a class of shot noise processes

1995 ◽  
Vol 32 (03) ◽  
pp. 707-726 ◽  
Author(s):  
Patrick Homble ◽  
William P. McCormick
Keyword(s):  
Poisson Process ◽  
Shot Noise ◽  
Impulse Response Function ◽  
Weak Limit ◽  
Intensity Function ◽  
Important Class ◽  
Homogeneous Poisson Process ◽  
Periodic Intensity ◽  
Non Homogeneous Poisson Process ◽  
Over Time

Shot noise processes form an important class of stochastic processes modeling phenomena which occur as shocks to a system and with effects that diminish over time. In this paper we present extreme value results for two cases — a homogeneous Poisson process of shocks and a non-homogeneous Poisson process with periodic intensity function. Shocks occur with a random amplitude having either a gamma or Weibull density and dissipate via a compactly supported impulse response function. This work continues work of Hsing and Teugels (1989) and Doney and O'Brien (1991) to the case of random amplitudes.

On multiple change-point estimation for Poisson process

2017 ◽  
Vol 47 (5) ◽  
pp. 1215-1233 ◽  
Author(s):  
O. V. Chernoyarov ◽  
Yu. A. Kutoyants ◽  
A. Top
Keyword(s):  
Poisson Process ◽  
Change Point ◽  
Point Estimation ◽  
Change Point Estimation
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