Hostname: page-component-f554764f5-nt87m Total loading time: 0 Render date: 2025-04-09T17:41:09.764Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimation of entropy for Poisson marked point processes

Part of: Stochastic processes Inference from stochastic processes Limit theorems

Published online by Cambridge University Press:  17 March 2017

P. Alonso-Ruiz  and
E. Spodarev
P. Alonso-Ruiz*
Affiliation:
Ulm University
E. Spodarev*
Affiliation:
Ulm University
*
* Current address: Department of Mathematics, University of Connecticut, 341 Mansfield Road, Unit 1009, Storrs, CT 06269-1009, USA. Email address: [email protected]
** Postal address: Ulm University, Helmholtzstr. 18, 89081 Ulm, Germany. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper a kernel estimator of the differential entropy of the mark distribution of a homogeneous Poisson marked point process is proposed. The marks have an absolutely continuous distribution on a compact Riemannian manifold without boundary. We investigate L2 and the almost surely consistency of this estimator as well as its asymptotic normality.

Keywords

Marked point processkernel density estimatorentropycentral limit theoremBoolean model

MSC classification

Primary: 62M30: Spatial processes
Secondary: 60G60: Random fields 60F05: Central limit and other weak theorems
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 1 , March 2017 , pp. 258 - 278
Copyright
Copyright © Applied Probability Trust 2017 

References

[1] Ahmad, I. A.and Lin, P. E. (1976).A nonparametric estimation of the entropy for absolutely continuous distributions.IEEE Trans. Inf. TheoryIT-22,372–375.CrossRefGoogle Scholar
[2] Bai, Z. D.,Radhakrishna Rao, C. and Zhao, L. C. (1988).Kernel estimators of density function of directional data.J. Multivariate Anal. 27,24–39.CrossRefGoogle Scholar
[3] Beirlant, J.,Dudewicz, E. J.,Györfi, L. and van der Meulen, E. C. (1997).Nonparametric entropy estimation: an overview.Int. J. Math. Statist. Sci. 6,17–39.Google Scholar
[4] Besse, A. L. (1978).Manifolds All of Whose Geodesics are Closed(Results Math. Rel. Areas 93).Springer,Berlin.CrossRefGoogle Scholar
[5] Boothby, W. M. (1986).An Introduction to Differentiable Manifolds and Riemannian Geometry(Pure App. Math. 120),2nd edn.Academic Press,Orlando, FL.Google Scholar
[6] Bulinski, A. and Shashkin, A. (2007).Limit Theorems for Associated Random Fields and Related Systems(Adv. Ser. Statist. Sci. App. Prob. 10).World Scientific Publishing,Hackensack, NJ.CrossRefGoogle Scholar
[7] Burton, R. and Waymire, E. (1985).Scaling limits for associated random measures.Ann. Prob. 13,1267–1278.CrossRefGoogle Scholar
[8] Chen, L. H. Y. and Shao, Q.-M. (2004).Normal approximation under local dependence.Ann. Prob. 32,1985–2028.CrossRefGoogle Scholar
[9] El Machkouri, M. (2014).Kernel density estimation for stationary random fields.ALEA Lat. Amer. J. Prob. Math. Statist. 11,259–279.Google Scholar
[10] Hall, P.,Watson, G. S. and Cabrera, J. (1987).Kernel density estimation with spherical data.Biometrika 74,751–762.CrossRefGoogle Scholar
[11] Heinrich, L. (1988).Asymptotic behaviour of an empirical nearest-neighbour distance function for stationary Poisson cluster processes.Math. Nachr. 136,131–148.CrossRefGoogle Scholar
[12] Heinrich, L.,Klein, S. and Moser, M. (2014).Empirical Mark covariance and product density function of stationary marked point processes–a survey on asymptotic results.Methodology. Comput. Appl. Prob. 16,283–293.CrossRefGoogle Scholar
[13] Heinrich, L.,Lück, S. and Schmidt, V. (2014).Asymptotic goodness-of-fit tests for the Palm mark distribution of stationary point processes with correlated marks.Bernoulli 20,1673–1697.CrossRefGoogle Scholar
[14] Henry, G. and Rodriguez, D. (2009).Kernel density estimation on Riemannian manifolds: asymptotic results.J. Math. Imaging Vision 34,235–239.CrossRefGoogle Scholar
[15] Kiderlen, M. (2001).Non-parametric estimation of the directional distribution of stationary line and fibre processes.Adv. Appl. Prob. 33,6–24.CrossRefGoogle Scholar
[16] Parzen, E. (1962).On estimation of a probability density function and mode.Ann. Math. Statist. 33,1065–1076.CrossRefGoogle Scholar
[17] Pawlas, Z. (2009).Empirical distributions in marked point processes.Stoch. Process. App. 119,4194–4209.CrossRefGoogle Scholar
[18] Pelletier, B. (2005).Kernel density estimation on Riemannian manifolds.Statist. Prob. Lett. 73,297–304.CrossRefGoogle Scholar
[19] Penrose, M. D. and Yukich, J. E. (2013).Limit theory for point processes in manifolds.Ann. Appl. Prob. 23,2161–2211.CrossRefGoogle Scholar
[20] Rosén, B. (1969).A note on asymptotic normality of sums of higher-dimensionally indexed random variables.Ark. Mat. 8,33–43.CrossRefGoogle Scholar
[21] Rosenblatt, M. (1956).A central limit theorem and a strong mixing condition.Proc. Nat. Acad. Sci. USA 42,43–47.CrossRefGoogle Scholar
[22] Sakai, T. (1996).Riemannian Geometry(Transl. Math. Monogr. 149).American Mathematical Society,Providence, RI.CrossRefGoogle Scholar
[23] Schuster, T. (2007).The Method of Approximate Inverse: Theory and Applications(Lecture Notes Math. 1906).Springer,Berlin.CrossRefGoogle Scholar
[24] Shannon, C. E. (1948).A mathematical theory of communication.Bell System Tech. J. 27,379–423,623–656.CrossRefGoogle Scholar
[25] Spodarev, E.(ed.) (2013).Stochastic Geometry, Spatial Statistics and Random Fields(Lecture Notes Math. 2068).Springer,Heidelberg.CrossRefGoogle Scholar
[26] Stoyan, D.,Kendall, W. S. and Mecke, J. (1987).Stochastic Geometry and Its Applications.John Wiley,Chichester.Google Scholar
[27] Tsybakov, A. B. (2009).Introduction to Nonparametric Estimation.Springer,New York.CrossRefGoogle Scholar