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On the MLE for a spatial point pattern

Published online by Cambridge University Press:  01 July 2016

Francisco Montes  and
Jorge Mateu
Francisco Montes
Affiliation:
Universitat de València
Jorge Mateu
Affiliation:
Universitat Jaume 1
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Extract

Parameter estimation for a two-dimensional point pattern is difficult because most of the available stochastic models have intractable likelihoods ([2]). An exception is the class of Gibbs or Markov point processes ([1], [5]), where the likelihood typically forms an exponential family and is given explicitly up to a normalising constant. However, the latter is not known analytically, so parameter estimates must be based on approximations ([3], [6], [7]). In this paper we present comparisons amongst the different techniques available in the literature to obtain an approximation of the maximum likelihood estimate (MLE). Two stochastic methods are specifically illustrated: a Newton-Raphson algorithm ([7]) and the Robbins-Monro procedure ([8]). We use a very simple point process model, the Strauss process ([4]), to test and compare those approximations.

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 28 , Issue 2 , June 1996 , pp. 339
Copyright
Copyright © Applied Probability Trust 1996 

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References

[1] Baddeley, A. J. and Moller, J. (1989) Nearest-neighbour Markov point processes and random sets. Int. Statist. Rev. 57, 89–121.CrossRefGoogle Scholar
[2] Diggle, P. J. (1983) Statistical Analysis of Spatial Point Patterns. Academic Press, London.Google Scholar
[3] Diggle, P. J. and Gratton, R. J. (1984) Monte Carlo methods of inference for implicit statistical models (with discussion). J. R. Statist. Soc. B 46, 193–227.Google Scholar
[4] Kelly, F. P. and Ripley, B. D. (1976) On Strauss's model for clustering. Biometrika 63, 357–360.CrossRefGoogle Scholar
[5] Moyeed, R. A. and Baddeley, A. J. (1991) Stochastic approximation of the MLE for a spatial point pattern. Scand. J. Statist. 18, 39–50.Google Scholar
[6] Ogata, Y. and Tanemura, M. (1984) Likelihood analysis of spatial point patterns. J. R. Statist. Soc. B 46, 496–518.Google Scholar
[7] Penttinen, A. (1984) Modelling interactions in spatial point patterns: parameter estimation by the maximum likelihood method. Jyväskylä Studies in Computer Science. (Economics and Statistics 7).Google Scholar
[8] Robbins, H. and Monro, S. (1951) A stochastic approximation method. Ann. Math. Statist. 22, 400–407.CrossRefGoogle Scholar