Hostname: page-component-89b8bd64d-dvtzq Total loading time: 0 Render date: 2026-05-08T16:35:10.950Z Has data issue: false hasContentIssue false
  • English
  • Français

A poisson process whose rate is a hidden Markov process

Published online by Cambridge University Press:  01 July 2016

D. S. Freed  and
L. A. Shepp
D. S. Freed
Affiliation:
Harvard University
L. A. Shepp*
Affiliation:
Bell Laboratories
*
∗∗ Postal address: Bell Laboratories, 600 Mountain Ave, Murray Hill, NJ 07924, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let a Poisson process be observed whose output rate is one of two levels given by the state of an unseen Markov process. If one of the levels is 0, a simple formula is given for the best guess of the state at any instant based on the stream of past Poisson events. In other cases bounds are given for the likelihood ratio of the state probabilities given the event stream.

Keywords

POISSONCOXDOUBLY STOCHASTICHIDDEN RATELIKELIHOOD RATIO

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 14 , Issue 1 , March 1982 , pp. 21 - 36
Copyright
Copyright © Applied Probability Trust 1982 

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.)

Article purchase

Temporarily unavailable

Footnotes

a

Present address: 970 Evans Hall, Mathematics Department, University of California, Berkeley, CA 94720, U.S.A.

Research carried out while this author was a summer employee at Bell Laboratories.

References

1. Boel, K. and Benes, V. E. (1981) Recursive nonlinear estimation of a diffusion acting as the rate of an observed Poisson process. IEEE Trans. Information Theory. CrossRefGoogle Scholar
2. Cox, D. R. (1955) Some statistical methods related with series of events. J. R. Statist. Soc. B 17, 129–164.Google Scholar
3. Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
4. Grandell, J. (1976) Doubly Stochastic Poisson Processes. Lecture Notes in Mathematics 529, Springer-Verlag, Berlin.CrossRefGoogle Scholar
5. Kingman, J. F. C. (1964) On doubly stochastic Poisson processes. Proc. Camb. Phil. Soc. 60, 923–930.CrossRefGoogle Scholar
6. Lipster, R. Sh. and Shiryayev, A. N. (1978) Statistics of Random Processes II. Springer-Verlag, Berlin.Google Scholar
7. Rudemo, M. E. (1975) Prediction and smoothing for partially observed Markov chains. J. Math. Anal. Appl. 49, 1–23.CrossRefGoogle Scholar
8. Snyder, D. L. (1975) Random Point Processes. Wiley, New York.Google Scholar