Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-05-20T23:03:59.363Z Has data issue: false hasContentIssue false

Estimation of entropy for Poisson marked point processes

Published online by Cambridge University Press:  17 March 2017

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: patricia.alonso-ruiz@uconn.edu
** Postal address: Ulm University, Helmholtzstr. 18, 89081 Ulm, Germany. Email address: evgeny.spodarev@uni-ulm.de
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.

Type
Research Article
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-22,372375.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. 27,2439.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. 6,1739.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. 13,12671278.CrossRefGoogle Scholar
[8] Chen, L. H. Y. and Shao, Q.-M. (2004).Normal approximation under local dependence.Ann. Prob. 32,19852028.CrossRefGoogle Scholar
[9] El Machkouri, M. (2014).Kernel density estimation for stationary random fields.ALEA Lat. Amer. J. Prob. Math. Statist. 11,259279.Google Scholar
[10] Hall, P.,Watson, G. S. and Cabrera, J. (1987).Kernel density estimation with spherical data.Biometrika 74,751762.CrossRefGoogle Scholar
[11] Heinrich, L. (1988).Asymptotic behaviour of an empirical nearest-neighbour distance function for stationary Poisson cluster processes.Math. Nachr. 136,131148.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. 16,283293.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 20,16731697.CrossRefGoogle Scholar
[14] Henry, G. and Rodriguez, D. (2009).Kernel density estimation on Riemannian manifolds: asymptotic results.J. Math. Imaging Vision 34,235239.CrossRefGoogle Scholar
[15] Kiderlen, M. (2001).Non-parametric estimation of the directional distribution of stationary line and fibre processes.Adv. Appl. Prob. 33,624.CrossRefGoogle Scholar
[16] Parzen, E. (1962).On estimation of a probability density function and mode.Ann. Math. Statist. 33,10651076.CrossRefGoogle Scholar
[17] Pawlas, Z. (2009).Empirical distributions in marked point processes.Stoch. Process. App. 119,41944209.CrossRefGoogle Scholar
[18] Pelletier, B. (2005).Kernel density estimation on Riemannian manifolds.Statist. Prob. Lett. 73,297304.CrossRefGoogle Scholar
[19] Penrose, M. D. and Yukich, J. E. (2013).Limit theory for point processes in manifolds.Ann. Appl. Prob. 23,21612211.CrossRefGoogle Scholar
[20] Rosén, B. (1969).A note on asymptotic normality of sums of higher-dimensionally indexed random variables.Ark. Mat. 8,3343.CrossRefGoogle Scholar
[21] Rosenblatt, M. (1956).A central limit theorem and a strong mixing condition.Proc. Nat. Acad. Sci. USA 42,4347.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. 27,379423,623656.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