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A bivariate Levy process with negative binomial and gamma marginals

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Author

Summary, in English

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.

Publishing year

2008

Language

English

Pages

1418-1437

Publication/Series

Journal of Multivariate Analysis

Volume

99

Issue

7

Document type

Journal article

Publisher

Academic Press

Topic

  • Probability Theory and Statistics

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

Status

Published

ISBN/ISSN/Other

  • ISSN: 0047-259X