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Distributional properties of the negative binomial Lévy process

Kozubowski, Tomasz and Podgorski, Krzysztof LU (2009) In Probability and Mathematical Statistics 29(Fasc. 1). p.43-71
Abstract
The geometric distribution leads to a Lévy process parameterized

by the probability of success. The resulting negative binomial process

(NBP) is a purely jump and non-decreasing process with general negative

binomial marginal distributions. We review various stochastic mechanisms

leading to this process, and study its distributional structure. These

results enable us to establish strong convergence of the NBP in the supremum

norm to the gamma process, and lead to a straightforward algorithm

for simulating sample paths.We also include a brief discussion of estimation

of the NPB parameters, and present an example from hydrology illustrating

possible applications... (More)
The geometric distribution leads to a Lévy process parameterized

by the probability of success. The resulting negative binomial process

(NBP) is a purely jump and non-decreasing process with general negative

binomial marginal distributions. We review various stochastic mechanisms

leading to this process, and study its distributional structure. These

results enable us to establish strong convergence of the NBP in the supremum

norm to the gamma process, and lead to a straightforward algorithm

for simulating sample paths.We also include a brief discussion of estimation

of the NPB parameters, and present an example from hydrology illustrating

possible applications of this model. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Borehole data, Cluster Poisson process, Compound Poisson process: Count data: Cox process, Discrete Lévy process, Doubly stochastic Poisson process, Fractures, Gamma-Poisson process, Gamma process: Geometric distribution, Immigration birth process, Infinite divisibility, Logarithmic distribution: Over-dispersion, Pascal distribution, Point process, Random time transformation, Subordination, Simulation
in
Probability and Mathematical Statistics
volume
29
issue
Fasc. 1
pages
43 - 71
publisher
Center for Probability and Mathematical Statistics, Wroclaw
ISSN
0208-4147
language
English
LU publication?
yes
id
226dbe23-3c63-4a43-aeea-776b7c7349a9 (old id 3049633)
alternative location
http://www.math.uni.wroc.pl/~pms/files/29.1/Article/29.1.3.pdf
date added to LUP
2012-09-10 12:39:08
date last changed
2016-04-15 21:09:34
@article{226dbe23-3c63-4a43-aeea-776b7c7349a9,
  abstract     = {The geometric distribution leads to a Lévy process parameterized<br/><br>
by the probability of success. The resulting negative binomial process<br/><br>
(NBP) is a purely jump and non-decreasing process with general negative<br/><br>
binomial marginal distributions. We review various stochastic mechanisms<br/><br>
leading to this process, and study its distributional structure. These<br/><br>
results enable us to establish strong convergence of the NBP in the supremum<br/><br>
norm to the gamma process, and lead to a straightforward algorithm<br/><br>
for simulating sample paths.We also include a brief discussion of estimation<br/><br>
of the NPB parameters, and present an example from hydrology illustrating<br/><br>
possible applications of this model.},
  author       = {Kozubowski, Tomasz and Podgorski, Krzysztof},
  issn         = {0208-4147},
  keyword      = {Borehole data,Cluster Poisson process,Compound Poisson process: Count data: Cox process,Discrete Lévy process,Doubly stochastic Poisson process,Fractures,Gamma-Poisson process,Gamma process: Geometric distribution,Immigration birth process,Infinite divisibility,Logarithmic distribution: Over-dispersion,Pascal distribution,Point process,Random time transformation,Subordination,Simulation},
  language     = {eng},
  number       = {Fasc. 1},
  pages        = {43--71},
  publisher    = {Center for Probability and Mathematical Statistics, Wroclaw},
  series       = {Probability and Mathematical Statistics},
  title        = {Distributional properties of the negative binomial Lévy process},
  volume       = {29},
  year         = {2009},
}