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Mean and Variance of Vacancy for Hard‐Core Disc Processes and Applications
Author(s) -
Bondesson Lennart,
Fahlén Jessica
Publication year - 2003
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/1467-9469.00365
Subject(s) - mathematics , point process , dimension (graph theory) , variance (accounting) , torus , monte carlo method , markov chain , radius , point (geometry) , markov chain monte carlo , combinatorics , core (optical fiber) , statistical physics , geometry , statistics , physics , computer science , accounting , computer security , business , optics
. Hard‐core Strauss disc processes with inhibition distance r and disc radius R are considered. The points in the Strauss point process are thought of as trees and the discs as crowns. Formulas for the mean and the variance of the vacancy (non‐covered area) are derived. This is done both for the case of a fixed number of points and for the case of a random number of points. For tractability, the region is assumed to be a torus or, in one dimension, a circle in which case the discs are segments. In the one‐dimensional case, the formulas are exact for all r . This case, although less important in practice than the two‐dimensional case, has provided a lot of inspiration. In the two‐dimensional case, the formulas are only approximate but rather accurate for r < R . Markov Chain Monte Carlo simulations confirm that they work well. For R ≤ r < 2 R , no formulas are presented. A forestry estimation problem, which has motivated the research, is briefly considered as well as another application in spatial statistics.