On an optimal selection problem of Cowan and Zabczyk

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.

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Misspecified change-point estimation problem for a Poisson process

Journal of Applied Probability ◽

10.1239/jap/1085496596 ◽

2001 ◽

Vol 38 (A) ◽

pp. 122-130 ◽

Cited By ~ 4

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∗.

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Kekonvergenan MSE Penduga Kernel Seragam Fungsi Intensitas Proses Poisson Periodik dengan Tren Fungsi Pangkat

ComTech Computer Mathematics and Engineering Applications ◽

10.21512/comtech.v3i1.2403 ◽

2012 ◽

Vol 3 (1) ◽

pp. 204

Author(s):

Ro’fah Nur Rachmawati

Keyword(s):

Stochastic Process ◽

Poisson Process ◽

Continuous Time ◽

Kernel Estimator ◽

Intensity Function ◽

Rank Function ◽

Service Center ◽

Discrete State ◽

Number Of Customers ◽

Discrete State Space

The number of customers who come to a service center will be different for each particular time. However, it can be modeled by a stochastic process. One particular form of stochastic process with continuous time and discrete state space is a periodic Poisson process. The intensity function of the process is generally unknown, so we need a method to estimate it. In this paper an estimator of kernel uniform of a periodic Poisson process is formulated with a trend component in a rank function (rank coefficient 0 <b <1 is known, and the slope coefficient of the power function (trend) a> 0 is known). It is also demonstrated the convergenity of the estimators obtained. The result of this paper is a formulation of a uniform kernel estimator for the intensity function of a periodic Poisson process with rank function trends (for the case “a” is known) and the convergenity proof of the estimators obtained.

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

Journal of Mathematical Sciences ◽

10.1007/bf02368732 ◽

1995 ◽

Vol 75 (2) ◽

pp. 1504-1509 ◽

Cited By ~ 1

Author(s):

V. V. Dovgalyuk

Keyword(s):

Poisson Process ◽

Limit Theorems ◽

Intensity Function ◽

Kernel Estimate ◽

Inhomogeneous Poisson Process

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

Canadian Journal of Statistics ◽

10.1002/cjs.11514 ◽

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

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Misspecified change-point estimation problem for a Poisson process

Journal of Applied Probability ◽

10.1017/s0021900200112720 ◽

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 ∗.

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Poisson processes directed by subordinators, stuttering poisson and pseudo-poisson processes, with applications to actuarial mathematics

Journal of Physics Conference Series ◽

10.1088/1742-6596/2131/2/022107 ◽

2021 ◽

Vol 2131 (2) ◽

pp. 022107

Author(s):

O Rusakov ◽

Yu Yakubovich

Keyword(s):

Poisson Process ◽

Continuous Time ◽

Random Variables ◽

Compound Poisson Process ◽

Poisson Processes ◽

Finite Variance ◽

Uhlenbeck Process ◽

Skorokhod Space ◽

Actuarial Mathematics ◽

Ornstein Uhlenbeck Process

Abstract Weconsider a PSI-process, that is a sequence of random variables (&), i = 0.1,…, which is subordinated by a continuous-time non-decreasing integer-valued process N(t): <K0 = ÇN(ty Our main example is when /V(t) itself is obtained as a subordination of the standard Poisson process 77(s) by a non-decreasing Lévy process S(t): N(t) = 77(S(t)).The values (&)one interprets as random claims, while their accumulated intensity S(t) is itself random. We show that in this case the process 7V(t) is a compound Poisson process of the stuttering type and its rate depends just on the value of theLaplace exponent of S(t) at 1. Under the assumption that the driven sequence (&) consists of i.i.d. random variables with finite variance we calculate a correlation function of the constructed PSI-process. Finally, we show that properly rescaled sums of such processes converge to the Ornstein-Uhlenbeck process in the Skorokhod space. We suppose that the results stated in the paper mightbe interesting for theorists and practitioners in insurance, in particular, for solution of reinsurance tasks.

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Some Results on Interarrival Times of Nonhomogeneous Poisson Processes

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800004186 ◽

1996 ◽

Vol 10 (1) ◽

pp. 75-85 ◽

Cited By ~ 11

Author(s):

Subhash C. Kochar

Keyword(s):

Poisson Process ◽

Stochastic Order ◽

Stochastic Ordering ◽

Intensity Function ◽

Poisson Processes ◽

Constant Intensity ◽

Nonhomogeneous Poisson Processes ◽

Interarrival Times ◽

Hazard Rate Ordering ◽

Intensity Functions

It is well known that in the case of a Poisson process with constant intensity function the interarrival times are independent and identically distributed, each having exponential distribution. We study this problem when the intensity function is monotone. In particular, we show that in the case of a nonhomogeneous Poisson process with decreasing (increasing) intensity the interarrival times are increasing (decreasing) in the hazard rate ordering sense and they are also jointly likelihood ratio ordered (cf. Shanthikumar and Yao, 1991, Bivariate characterization of some stochastic order relations, Advances in Applied Probability 23: 642–659). This result is stronger than the usual stochastic ordering between the successive interarrival times. Also in this case, the interarrival times are conditionally increasing in sequence and, as a consequence, they are associated. We also consider the problem of comparing two nonhomogeneous Poisson processes in terms of the ratio of their intensity functions and establish some results on the successive number of events from one process occurring between two consecutive occurrences from the second process.

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