Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-06-24T02:17:55.458Z Has data issue: false hasContentIssue false
  • English
  • Français

A new graph related to the directions of nearest neighbours in a point process

Part of: Multivariate analysis Limit theorems Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

S. N. Chiu  and
I. S. Molchanov
S. N. Chiu*
Affiliation:
Hong Kong Baptist University
I. S. Molchanov*
Affiliation:
University of Bern
*
∗ Postal address: Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong. Email address: snchiu@math.hkbu.edu.hk
∗∗ Postal address: Department of Mathematical Statistics and Actuarial Science, University of Bern, Sidlerstr. 5, CH-3012 Bern, Switzerland.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper introduces a new graph constructed from a point process. The idea is to connect a point with its nearest neighbour, then to the second nearest and continue this process until the point belongs to the interior of the convex hull of these nearest neighbours. The number of such neighbours is called the degree of a point. We derive the distribution of the degree of the typical point in a Poisson process, prove a central limit theorem for the sum of degrees, and propose an edge-corrected estimator of the distribution of the degree that is unbiased for a stationary Poisson process. Simulation studies show that this degree is a useful concept that allows the separation of clustering and repulsive behaviour of point processes.

Keywords

Point processrandom graphconvex hulldegree

MSC classification

Primary: 60G55: Point processes 60D05: Geometric probability and stochastic geometry
Secondary: 60F05: Central limit and other weak theorems 62H11: Directional data; spatial statistics
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 1 , March 2003 , pp. 47 - 55
Copyright
Copyright © Applied Probability Trust 2003 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baddeley, A. J. and Silverman, B. W. (1984). A cautionary example on the use of second-order methods for analyzing point patterns. Biometrics 40, 1089–1093.Google Scholar
Bolthausen, E. (1982). On the central limit theorem for stationary mixing random fields. Ann. Prob. 10, 1047–1050.CrossRefGoogle Scholar
Chiu, S. N. (1999). An unbiased estimator for the survival function of censored data. Commun. Statist. Theory Meth. 28, 2249–2260.Google Scholar
Hall, P. (1988). Introduction to The Theory of Coverage Processes. John Wiley, New York.Google Scholar