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Likelihood and nearest-neighbor distance properties of multidimensional Poisson cluster processes

Published online by Cambridge University Press:  14 July 2016

Michel Baudin
Michel Baudin*
Affiliation:
École Nationale Supérieure des Mines de Paris
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Abstract

The probability generating functional representation of a multidimensional Poisson cluster process is utilized to derive a formula for its likelihood function, but the prohibitive complexity of this formula precludes its practical application to statistical inference. In the case of isotropic processes, it is however feasible to compute functions such as the probability Q(r) of finding no point in a disc of radius r and the probability Q(r | 0) of nearest-neighbor distances greater than r, as well as the expected number C(r | 0) of points at a distance less than r from a given point. Explicit formulas and asymptotic developments are derived for these functions in the n-dimensional case. These can effectively be used as tools for statistical analysis.

Keywords

POINT PROCESSESPOISSON CLUSTER PROCESSESPROBABILITY GENERATING FUNCTIONAL
Type
Research Papers
Information
Journal of Applied Probability , Volume 18 , Issue 4 , December 1981 , pp. 879 - 888
Copyright
Copyright © Applied Probability Trust 1981 

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References

Ammann, L. P. and Thall, P. F. (1979) Count distributions, orderliness and invariance of Poisson cluster processes. J. Appl. Prob. 16, 261–273.Google Scholar
Davies, R. B. (1977) Testing the hypothesis that a point process is Poisson. Adv. Appl. Prob. 9, 724–746.Google Scholar
Fisher, L. (1972) Mathematical theory of multidimensional point processes. In Stochastic Point Processes , ed. Lewis, P. A. W. Wiley, New York.Google Scholar
Kagan, Y. Y. and Knopoff, L. (1976a) Statistical search for non-random features of the seismicity of strong earthquakes. Phys. Earth, Planetary Interiors 12, 291–318.CrossRefGoogle Scholar
Kagan, Y. Y. and Knopoff, L. (1976b) Earthquake risk prediction as a stochastic process. Phys. Earth, Planetary Interiors 14, 97–108.Google Scholar
Macchi, O. (1975) The coincidence approach to stochastic point processes. Adv. Appl. Prob. 7, 83–122.Google Scholar
Matthes, K., Kerstan, J. and Mecke, J. (1975) Infinitely Divisible Point Processes. Wiley, New York.Google Scholar
Mecke, J. (1967) Stationäre zufällige Masse auf lokalkompakten, abelschen Gruppen. Z. Wahrscheinlichkeitsth. 9, 36–58.Google Scholar
Moyal, J. E. (1962) The general theory of stochastic population processes. Acta Math. 108, 1–31.Google Scholar
Neyman, J. and Scott, E. (1958) Statistical approach to problems of cosmology. J. R. Statist. Soc. B 20, 1–43.Google Scholar
Neyman, J. and Scott, E. (1972) Processes of clustering and applications. In Stochastic Point Processes , ed. Lewis, P. A. W., Wiley, New York.Google Scholar
Paloheimo, J. E. (1971) Discussion in Statistical Ecology 1, ed. Patil, G. P., Pielou, E. C. and Waters, W. E., Pennsylvania State University Press, 210–212.Google Scholar
Vere-Jones, D. (1968) Some applications of probability generating functionals to the study of input-output streams. J. R. Statist. Soc. B30, 321–333.Google Scholar
Vere-Jones, D. (1970) Stochastic models for earthquake occurrence. J. R. Statist. Soc. B 32, 1–62.Google Scholar
Vere-Jones, D. (1978) Space-time correlations for microearthquakes — a pilot study. Suppl. Adv. Appl. Prob. 10, 73–87.CrossRefGoogle Scholar
Warren, W. G. (1971) The center-satellite concept as a basis of ecological sampling. In Statistical Ecology 2, ed. Patil, G. P., Pielou, E. C. and Waters, W. E., Pennsylvania State University Press.Google Scholar
Westcott, M. (1972) The probability generating functional. J. Austral. Math. Soc. 14, 448–466.CrossRefGoogle Scholar