A bivariate Lévy process with negative binomial and gamma marginals
Tomas J. Kozubowski, Anna K. Panorska and Krzysztof Podgórski
Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,
2008
ISSN 14039338

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 whose coordinates are
correlated negative binomial and gamma processes. We derive basic properties
of this process, including its covariance structure, representations, and
stochastic selfsimilarity. We study its bivariate marginal distributions
of correlated gamma and negative binomial variables, and present their 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.




Key words:

discrete Lévy process, gamma process, gammaPoisson process, infinite
divisibility, maximum likelihood estimation, negative binomial process,
operational time, random summation, random time transformation, stability,
subordination, selfsimilarity

