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A bivariate Levy process with negative binomial and gamma marginals

Kozubowski, Tomasz J; Panorska, Anna K and Podgorski, Krzysztof LU (2008) In Journal of Multivariate Analysis 99(7). p.1418-1437
Abstract
The joint distribution of X and N, where N has a geometric distribution and X is the sum of N IID exponential variables (independent of N), is infinitely divisible. This leads to a bivariate Levy process {(X(t), N(t)), t >= 0}, whose coordinates are correlated negative binomial and gamma processes. We derive basic properties of this process, including its covariance structure, representations, and stochastic self-similarity. We examine the joint distribution of (X(t), N(t)) at a fixed time t, along with the marginal and conditional distributions, joint integral transforms, moments, infinite divisibility, and stability with respect to random summation. We also discuss maximum likelihood estimation and simulation for this model.
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
operational time, random summation, random time transformation, stability, subordination self-similarity, negative binomial process, maximum likelihood estimation, divisibility, infinite, gamma Poisson process, discrete Levy process, gamma process
in
Journal of Multivariate Analysis
volume
99
issue
7
pages
1418 - 1437
publisher
Academic Press
external identifiers
  • wos:000257221700005
  • scopus:45049087244
ISSN
0047-259X
DOI
10.1016/j.jmva.2008.02.029
language
English
LU publication?
yes
id
8178ad0b-9f3a-4a1d-a6d7-51e6bfc3b442 (old id 1186763)
date added to LUP
2008-09-03 16:34:24
date last changed
2017-01-01 05:54:10
@article{8178ad0b-9f3a-4a1d-a6d7-51e6bfc3b442,
  abstract     = {The joint distribution of X and N, where N has a geometric distribution and X is the sum of N IID exponential variables (independent of N), is infinitely divisible. This leads to a bivariate Levy process {(X(t), N(t)), t >= 0}, whose coordinates are correlated negative binomial and gamma processes. We derive basic properties of this process, including its covariance structure, representations, and stochastic self-similarity. We examine the joint distribution of (X(t), N(t)) at a fixed time t, along with the marginal and conditional distributions, joint integral transforms, moments, infinite divisibility, and stability with respect to random summation. We also discuss maximum likelihood estimation and simulation for this model.},
  author       = {Kozubowski, Tomasz J and Panorska, Anna K and Podgorski, Krzysztof},
  issn         = {0047-259X},
  keyword      = {operational time,random summation,random time transformation,stability,subordination self-similarity,negative binomial process,maximum likelihood estimation,divisibility,infinite,gamma Poisson process,discrete Levy process,gamma process},
  language     = {eng},
  number       = {7},
  pages        = {1418--1437},
  publisher    = {Academic Press},
  series       = {Journal of Multivariate Analysis},
  title        = {A bivariate Levy process with negative binomial and gamma marginals},
  url          = {http://dx.doi.org/10.1016/j.jmva.2008.02.029},
  volume       = {99},
  year         = {2008},
}