Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-09T06:05:51.477Z Has data issue: false hasContentIssue false
  • English
  • Français

On Updating Algorithms and Inference for Stochastic Point Processes

Published online by Cambridge University Press:  05 September 2017

D. Vere-Jones
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is an attempt to interpret and extend, in a more statistical setting, techniques developed by D. L. Snyder and others for estimation and filtering for doubly stochastic point processes. The approach is similar to the Kalman-Bucy approach in that the updating algorithms can be derived from a Bayesian argument, and lead ultimately to equations which are similar to those occurring in stochastic approximation theory. In this paper the estimates are derived from a general updating formula valid for any point process. It is shown that almost identical formulae arise from updating the maximum likelihood estimates, and on this basis it is suggested that in practical situations the sequence of estimates will be consistent and asymptotically efficient. Specific algorithms are derived for estimating the parameters in a doubly stochastic process in which the rate alternates between two levels.

Type
Part VI — Statistical Inference on Stochastic Processes
Information
Journal of Applied Probability , Volume 12 , Issue S1 , 1975 , pp. 239 - 259
Copyright
Copyright © 1975 Applied Probability Trust 

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

Albert, A. E. and Gardner, L. A. (Jr.) (1967) Stochastic Approximation and Nonlinear Regression. M.I.T. Press Research Monograph No. 42, Cambridge, Mass.Google Scholar
Brillinger, D. R. (1972) The spectral analysis of stationary interval functions. Proc. Sixth Berkeley Symp. Math. Statist. Prob. 1, 483–513.Google Scholar
Daley, D. J. (1974) Various concepts of orderliness for point-processes. Stochastic Geometry. (Ed. Harding, E. F. and Kendall, D. G.), Wiley, London, 148–161.Google Scholar
Davies, R. B. (1973) Asymptotic inference on stationary Gaussian time-series,. Adv. Appl. Prob. 5, 469–497.Google Scholar
Forrester, R. H. (Jr) and Snyder, D. L. (1973) Phasetracking performance of direct detection optical receivers. Preprint from Biomedical Computer Laboratory, Washington University, St. Louis.Google Scholar
Jowett, J. H. (1974) The a posteriori distribution of the rate function of a doubly stochastic Poisson process. Preprint from Mathematics Department, Victoria University, Wellington.Google Scholar
Cam, L. M. Le (1960) Locally asymptotically normal families of distributions. Univ. California Publ. Statist. 3, 37–98.Google Scholar
Cam, L. M. Le (1969) Théorie Asymptotique de la Décision Statistique. Montreal University Press, Montreal.Google Scholar
Papangelou, F. (1974) On the palm probabilities of processes of points and processes of lines. Stochastic Geometry. (Ed. Harding, E. F. and Kendall, D. G.) Wiley, London, 114–147.Google Scholar
Rudemo, M. (1973) Point process generated by transitions of Markov chains. Adv. Appl. Prob. 5, 262–286.Google Scholar
Snyder, D. L. (1972a) Filtering and detection for doubly stochastic Poisson processes. IEEE Trans. Information Theory IT–18 91–102.Google Scholar
Snyder, D. L. (1972b) Smoothing for doubly stochastic Poisson processes. IEEE Trans. Information Theory IT–18 558–662.CrossRefGoogle Scholar
Snyder, D. L. (1973) Information processing for observed jump processes. Information and Control 22, 69–78.CrossRefGoogle Scholar
Snyder, D. L. and Rhodes, I. B. (1972) Phase and frequency tracking in direct detection optical communications systems. IEEE Trans. Communications COM–20,1139–1142.Google Scholar
Yashin, A. I. (1970) Filtering of jump processes. Avtomat. i Telemeh. 5, 52–58.Google Scholar