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Generalized Gamma measures and shot-noise Cox processes

Part of: Probabilistic methods, simulation and stochastic differential equations Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Anders Brix
Anders Brix*
Affiliation:
Royal Veterinary and Agricultural University
*
∗ Postal address: Guy Carpenter Instrat, Aldgate House, 6th Floor, 33 Aldgate High Street, London EC3N 1AQ. Email address: [email protected]
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Abstract

A parametric family of completely random measures, which includes gamma random measures, positive stable random measures as well as inverse Gaussian measures, is defined. In order to develop models for clustered point patterns with dependencies between points, the family is used in a shot-noise construction as intensity measures for Cox processes. The resulting Cox processes are of Poisson cluster process type and include Poisson processes and ordinary Neyman-Scott processes.

We show characteristics of the completely random measures, illustrated by simulations, and derive moment and mixing properties for the shot-noise random measures. Finally statistical inference for shot-noise Cox processes is considered and some results on nearest-neighbour Markov properties are given.

Keywords

Spatial point processrandom measurestable measuregamma measuregamma processshot-noise processCox processPoisson cluster processNeyman-Scott process60G57

MSC classification

Primary: 60G57: Random measures
Secondary: 60G60: Random fields 65C05: Monte Carlo methods
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 31 , Issue 4 , December 1999 , pp. 929 - 953
Copyright
Copyright © Applied Probability Trust 1999 

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