Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-26T00:05:13.305Z Has data issue: false hasContentIssue false
  • English
  • Français

Second-order approximation to the characteristic function of certain point-process integrals

Published online by Cambridge University Press:  01 July 2016

Steven P. Ellis
Steven P. Ellis*
Affiliation:
Massachusetts Institute of Technology
*
∗Present address: Department of Statistics, University of Rochester, Rochester, NY 14627, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

A general way to look at kernel estimates of densities is to regard them as stochastic integrals with respect to a spatial point process. Under regularity conditions these behave asymptotically as if the point process were Poisson. However, this Poisson approximation may not work well if the data exhibits a lot of clustering. In this paper a more refined approximation to the characteristic functions of the integrals is developed. For clustered data, a ‘Gauss–Poisson’ approximation works better than the Poisson.

Keywords

SPATIAL POINT PROCESSSTOCHASTIC INTEGRALFACTORIAL CUMULANT MEASURESDENSITY ESTIMATIONPALM DISTRIBUTIONGAUSS-POISSON PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 19 , Issue 3 , September 1987 , pp. 546 - 559
Copyright
Copyright © Applied Probability Trust 1987 

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.)

Footnotes

This work was partially supported by United States National Science Foundation Grants MCS 75–10376, PFR 79–01642, and MCS 82–02122.

References

Brillinger, D. R. (1978) A note on a representation for the Gauss-Poisson process. Stoch. Proc. Appl. 6, 135–137.Google Scholar
Ellis, S. P. (1981) Density Estimation for Point Process Data. Ph.D. Dissertation, Department of Statistics, University of California, Berkeley.Google Scholar
Ellis, S. P. (1983) Density estimation for multivariate data generated by a point process. Technical Report NSF 39, Statistics Center, Massachusetts Institute of Technology.Google Scholar
Ellis, S. P. (1986a) A limit theorem for spatial point processes. Adv. Appl. Prob. 18, 646–659.Google Scholar
Ellis, S. P. (1986) Density estimation for point processes. J. Multivariate Anal. (submitted).Google Scholar
Kallenberg, O. (1976) Random Measures. Academie-Verlag, Berlin; Academic Press, New York.Google Scholar
Vere-Jones, D. (1978) Space time correlations for microearthquakes–A pilot study. Suppl. Appl. Prob. 10, 73–87.Google Scholar
Westcott, M. (1972) The probability generating functional. J. Austral. Math. Soc. 14, 448–466.Google Scholar