Skip to main content
×
×
Home

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

  • Michel Baudin (a1)
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.

Copyright
References
Hide All
Ammann L. P. and Thall P. F. (1979) Count distributions, orderliness and invariance of Poisson cluster processes. J. Appl. Prob. 16, 261273.
Davies R. B. (1977) Testing the hypothesis that a point process is Poisson. Adv. Appl. Prob. 9, 724746.
Fisher L. (1972) Mathematical theory of multidimensional point processes. In Stochastic Point Processes, ed. Lewis P. A. W.Wiley, New York.
Kagan Y. Y. and Knopoff L. (1976a) Statistical search for non-random features of the seismicity of strong earthquakes. Phys. Earth, Planetary Interiors 12, 291318.
Kagan Y. Y. and Knopoff L. (1976b) Earthquake risk prediction as a stochastic process. Phys. Earth, Planetary Interiors 14, 97108.
Macchi O. (1975) The coincidence approach to stochastic point processes. Adv. Appl. Prob. 7, 83122.
Matthes K., Kerstan J. and Mecke J. (1975) Infinitely Divisible Point Processes. Wiley, New York.
Mecke J. (1967) Stationäre zufällige Masse auf lokalkompakten, abelschen Gruppen. Z. Wahrscheinlichkeitsth. 9, 3658.
Moyal J. E. (1962) The general theory of stochastic population processes. Acta Math. 108, 131.
Neyman J. and Scott E. (1958) Statistical approach to problems of cosmology. J. R. Statist. Soc. B 20, 143.
Neyman J. and Scott E. (1972) Processes of clustering and applications. In Stochastic Point Processes, ed. Lewis P. A. W., Wiley, New York.
Paloheimo J. E. (1971) Discussion in Statistical Ecology 1, ed. Patil G. P., Pielou E. C. and Waters W. E., Pennsylvania State University Press, 210212.
Vere-Jones D. (1968) Some applications of probability generating functionals to the study of input-output streams. J. R. Statist. Soc. B30, 321333.
Vere-Jones D. (1970) Stochastic models for earthquake occurrence. J. R. Statist. Soc. B 32, 162.
Vere-Jones D. (1978) Space-time correlations for microearthquakes — a pilot study. Suppl. Adv. Appl. Prob. 10, 7387.
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.
Westcott M. (1972) The probability generating functional. J. Austral. Math. Soc. 14, 448466.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 4 *
Loading metrics...

Abstract views

Total abstract views: 92 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 18th December 2017. This data will be updated every 24 hours.