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Poisson limits and nonparametric estimation for pairwise interaction point processes

Part of: Nonparametric inference Inference from stochastic processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Wei-Bin Chang  and
John A. Gubner
Wei-Bin Chang*
Affiliation:
University of Wisconsin-Madison
John A. Gubner*
Affiliation:
University of Wisconsin-Madison
*
∗Postal address: 58-1 Chung Shan Rd, Cholan, Miaoli 36901, Taiwan. Email address: wei-bin@entropy.ece.wisc.edu
∗∗Postal address: Department of Electrical and Computer Engineering, University of Wisconsin, Madison, WI 53706, USA. Email address: gubner@engr.wisc.edu
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Abstract

The distribution of the interpoint distance process of a sequence of pairwise interaction point processes is considered. It is shown that, if the interaction function is piecewise-continuous, then the sequence of interpoint distance processes converges weakly to an inhomogeneous Poisson process under certain sparseness conditions. Convergence of the expectation of the interpoint distance process to the mean of the limiting Poisson process is also established. This suggests a new nonparametric estimator for the interaction function if independent identically distributed samples of the point process are available.

Keywords

Pairwise interaction point processinteraction functionnonparametric estimation

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 62G07: Density estimation 62M30: Spatial processes
Type
Short Communications
Information
Journal of Applied Probability , Volume 37 , Issue 1 , March 2000 , pp. 252 - 260
Copyright
Copyright © 2000 by The Applied Probability Trust 

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Footnotes

This work was supported by the Office of Naval Research, Mathematical Sciences Division, under ONR Grant N00014–94–1–0366

References

Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Billingsley, P. (1968). Probability and Measure, 2nd edn. John Wiley, New York.Google Scholar
Brown, T. C., and Silverman, B. W. (1979). Rates of Poisson convergence for U-statistics. J. Appl. Prob. 16, 428–432.Google Scholar
Diggle, P. J., Gates, D. J., and Stibbard, A. (1987). A nonparametric estimator for pairwise interaction point processes. Biometrika 74, 763–770.Google Scholar
Diggle, P. J., Fiksel, T., Grabarnik, P., Ogata, Y., Stoyan, D., and Tanemura, M. (1994). On parameter estimation for pairwise interaction point processes. Int. Statist. Rev. 62, 99–117.Google Scholar
Geyer, C. J. and Möller, J. (1994). Simulation procedures and likelihood inference for spatial point processes. Scand. J. Statist. 21, 359–373.Google Scholar
Karr, A. F. (1986). Point Processes and their Statistical Inference. Marcel Dekker, New York.Google Scholar
Kelly, F. P., and Ripley, B. D. (1976). A note on Strauss's model for clustering. Biometrika 63, 357–360.CrossRefGoogle Scholar
Kryscio, R. J., and Saunders, R. (1983). On interpoint distances for planar Poisson cluster processes. J. Appl. Prob. 20, 513–528.Google Scholar
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
Ogata, Y., and Tanemura, M. (1981). Estimation of interaction potentials of spatial point patterns through the maximum likelihood procedure. Ann. Inst. Statist. Math. 33B, 315–338.CrossRefGoogle Scholar
Ogata, Y., and Tanemura, M. (1984). Likelihood analysis of spatial point patterns. J. Roy. Statist. Soc. B 46, 496–518.Google Scholar
Penttinen, A. (1984). Modelling interaction in spatial point patterns: parameter estimation by the maximum likelihood method. Jyväsklä Stud. Comput. Sci. Econ. Statist. 7, 1–107.Google Scholar
Ripley, B. D. (1979). Simulating spatial patterns: dependent samplers from a multivariate density. Appl. Statist. 28, 109–112.Google Scholar
Saunders, R., and Funk, G. M. (1977). Poisson limits for a clustering model of Strauss. J. Appl. Prob. 14, 776–784.Google Scholar
Saunders, R., Kryscio, R. J., and Funk, G. M. (1982). Poisson limits for a hard-core clustering model. Stoc. Proc. Appl. 12, 97–106.CrossRefGoogle Scholar
Shiryayev, A. N. (1984). Probability. Springer, New York.CrossRefGoogle Scholar
Silverman, B. W., and Brown, T. C. (1978). Short distances, flat triangles and Poisson limits. J. Appl. Prob. 15, 815–825.Google Scholar
Strauss, D. J. (1975). A model for clustering. Biometrika 62, 467–475.Google Scholar
Strauss, D. J. (1986). On a general class of models for interaction. SIAM Rev. 28, 513–527.Google Scholar