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.
Original language | English |
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Pages (from-to) | 1418-1437 |
Journal | Journal of Multivariate Analysis |
Volume | 99 |
Issue number | 7 |
DOIs | |
Publication status | Published - 2008 |
Subject classification (UKÄ)
- Probability Theory and Statistics
Free 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