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

  • Steven P. Ellis (a1)

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.

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Present address: Department of Statistics, University of Rochester, Rochester, NY 14627, USA.

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

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References

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Brillinger, D. R. (1978) A note on a representation for the Gauss-Poisson process. Stoch. Proc. Appl. 6, 135137.
Ellis, S. P. (1981) Density Estimation for Point Process Data. Ph.D. Dissertation, Department of Statistics, University of California, Berkeley.
Ellis, S. P. (1983) Density estimation for multivariate data generated by a point process. Technical Report NSF 39, Statistics Center, Massachusetts Institute of Technology.
Ellis, S. P. (1986a) A limit theorem for spatial point processes. Adv. Appl. Prob. 18, 646659.
Ellis, S. P. (1986) Density estimation for point processes. J. Multivariate Anal. (submitted).
Kallenberg, O. (1976) Random Measures. Academie-Verlag, Berlin; Academic Press, New York.
Vere-Jones, D. (1978) Space time correlations for microearthquakes–A pilot study. Suppl. Appl. Prob. 10, 7387.
Westcott, M. (1972) The probability generating functional. J. Austral. Math. Soc. 14, 448466.

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