Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-16T09:21:38.080Z Has data issue: false hasContentIssue false
  • English
  • Français

The expected first-passage time for a sufficient statistic arising in the Poisson disorder problem

Published online by Cambridge University Press:  14 July 2016

K. Wickwire
K. Wickwire*
Affiliation:
University of Birmingham
Get access Rights & Permissions [Opens in a new window]

Abstract

In the Poisson disorder problem the probability that the parameter of a Poisson process y, has increased by a constant amount, given observations of ys, s ≦ t, is a Markov process of mixed type with jumps of variable (state-dependent) magnitude superimposed upon a drift which satisfies an ordinary differential equation. Using a likelihood-ratio transformation, one can reduce the backward equation satisfied by the expected first-passage time to a constant level for the mixed process to a differential-difference equation with a constant retardation. We discuss a method for solving this equation and present some numerical results on its solution. The accuracy of some approximations which are easier to calculate is investigated.

Keywords

POISSON DISORDER PROBLEMEXPECTED FIRST-PASSAGE TIMESMIXED PROCESSESDIFFERENTIAL-DIFFERENCE EQUATIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 2 , June 1979 , pp. 274 - 286
Copyright
Copyright © Applied Probability Trust 1979 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Davis, M. H. A. (1976) On the Poisson disorder problem. In Proc. Conf. Math. Control Theory, Zakopane, Poland, 1974, 65–72.Google Scholar
Gal'chuk, L. I. and Rozovskii, B. L. (1971) The ‘disorder’ problem for a Poisson process. Theory Prob. Appl. 16, 712–716.Google Scholar
Gihman, I. I. and Skorohod, A. V. (1972) Stochastic Differential Equations. Springer-Verlag, Berlin.Google Scholar
Rudemo, M. (1971) Doubly stochastic Poisson processes and process control. Adv. Appl. Prob. 4, 318–338.Google Scholar
Širyaev, A. N. (1973) Statistical Sequential Analysis. Transl. Math. Monographs 38, A.M.S., Providence, R.I. Google Scholar
Tuckwell, H. C. (1975) Determination of the inter-spike times of neurons receiving randomly arriving post-synaptic potentials. Biol. Cybernet. 18, 225–237.Google Scholar