Limit theorems for global measures of the deviation of a kernel estimate of intensity function of an inhomogeneous Poisson process

1995 ◽  
Vol 75 (2) ◽  
pp. 1504-1509 ◽  
Author(s):  
V. V. Dovgalyuk
Keyword(s):  
Poisson Process ◽  
Limit Theorems ◽  
Intensity Function ◽  
Kernel Estimate ◽  
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.

Convergence to equilibrium in a traffic model with restricted passing

1979 ◽  
Vol 16 (4) ◽  
pp. 881-889 ◽  
Author(s):  
Hans Dieter Unkelbach
Keyword(s):  
Poisson Process ◽  
Limit Theorems ◽  
Point Processes ◽  
Road Traffic ◽  
Poisson Processes ◽  
Traffic Model ◽  
Cluster Point ◽  
Convergence To Equilibrium ◽  
Constant Intensity

A road traffic model with restricted passing, formulated by Newell (1966), is described by conditional cluster point processes and analytically handled by generating functionals of point processes.The traffic distributions in either space or time are in equilibrium, if the fast cars form a Poisson process with constant intensity combined with Poisson-distributed queues behind the slow cars (Brill (1971)). It is shown that this state of equilibrium is stable, which means that this state will be reached asymptotically for general initial traffic distributions. Furthermore the queues behind the slow cars dissolve asymptotically like independent Poisson processes with diminishing rate, also independent of the process of non-queuing cars. To get these results limit theorems for conditional cluster point processes are formulated.

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.

Limit theorems for processes of randomly displaced regular events

1976 ◽  
Vol 13 (03) ◽  
pp. 530-537
Author(s):  
P. S. Collings
Keyword(s):  
Poisson Process ◽  
Limit Theorems ◽  
Counting Process ◽  
Storage Process ◽  
Finite Dimensional

Two types of limit theorems are proved for processes of randomly displaced regular events. Firstly, as the displacements tend to infinity, the counting process is shown to converge weakly to a Poisson process and secondly, as the interval between events tends to zero, convergence of the finite-dimensional distributions of the associated storage process to a diffusion is proved.

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