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Doubly stochastic Poisson processes and process control

Published online by Cambridge University Press:  01 July 2016

Mats Rudemo
Mats Rudemo*
Affiliation:
Research Institute of National Defence, Stockholm
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Abstract

Consider a Poisson point process with an intensity parameter forming a Markov chain with continuous time and finite state space. A system of ordinary differential equations is derived for the conditional distribution of the Markov chain given observations of the point process. An estimate of the current intensity, optimal in the least-squares sense, is computed from this distribution. Applications to reliability and replacement theory are given. A special case with two states, corresponding to a process in control and out of control, is discussed at length. Adjustment rules, based on the conditional probability of the out of control state, are studied. Regarded as a function of time, this probability forms a Markov process with the unit interval as state space. For the distribution of this process, integro-differential equations are derived. They are used to compute the average long run cost of adjustment rules.

Keywords

POINT PROCESSSTATE ESTIMATIONFORWARD RECURRENCE TIMEBAYESIAN RELIABILITY PREDICTIONDETECTION OF CHANGE POINTREPLACEMENT
Type
Research Article
Information
Advances in Applied Probability , Volume 4 , Issue 2 , August 1972 , pp. 318 - 338
Copyright
Copyright © Applied Probability Trust 1972 

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