Hostname: page-component-7c8c6479df-24hb2 Total loading time: 0 Render date: 2024-03-28T15:55:28.376Z Has data issue: false hasContentIssue false

On the Sums of Compound Negative Binomial and Gamma Random Variables

Published online by Cambridge University Press:  14 July 2016

P. Vellaisamy*
Affiliation:
Indian Institute of Technology Bombay
N. S. Upadhye*
Affiliation:
Indian Institute of Technology Bombay
*
Postal address: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, 400 076, India.
Postal address: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, 400 076, India.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the convolution of compound negative binomial distributions with arbitrary parameters. The exact expression and also a random parameter representation are obtained. These results generalize some recent results in the literature. An application of these results to insurance mathematics is discussed. The sums of certain dependent compound Poisson variables are also studied. Using the connection between negative binomial and gamma distributions, we obtain a simple random parameter representation for the convolution of independent and weighted gamma variables with arbitrary parameters. Applications to the reliability of m-out-of-n:G systems and to the shortest path problem in graph theory are also discussed.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2009 

References

Dhaene, J. et al. (2003). On the computation of the capital multiplier in the Fortis credit economic capital model. Belg. Actuarial Bull. 3, 5057.Google Scholar
Diaconis, P. and Perlman, M. D. (1990). Bounds for tail probabilities of weighted sums of independent gamma random variables. In Topics In Statistical Dependence (IMS Lecture Notes Monogr. Ser. 16), Institute of Mathematical Statistics, Hayward, CA, pp. 147166.Google Scholar
Drekic, S. and Willmot, G. E. (2005). On the moments of the time of ruin with applications to phase-type claims. N. Amer. Actuarial J. 9, 1730.CrossRefGoogle Scholar
Engel, J. and Zijlstra, M. (1980). A characterization of the gamma distribution by the negative binomial distribution. J. Appl. Prob. 17, 11381144.Google Scholar
Furman, E. (2007). On the convolution of negative binomial random variables. Statist. Prob. Lett. 77, 169172.CrossRefGoogle Scholar
Gundlach, M. and Lehrbass, F. (2004). CreditRisk{+ in the Banking Industry}. Springer, New York.CrossRefGoogle Scholar
Johnson, N. L., Kemp, A. W. and Kotz, S. (2005). Univariate Discrete Distributions, 3rd edn. John Wiley, Hoboken, NJ.CrossRefGoogle Scholar
Mieghen, P. V. (2006). Performance Analysis of Communication Networks and Systems. Cambridge University Press.Google Scholar
Panjer, H. H. and Willmot, G. E. (1981). Finite sum evaluation of negative binomial-exponential model. ASTIN Bull. 12, 133137.CrossRefGoogle Scholar
Scheuer, E. M. (1988). Reliability of an m-out-of-n system when component failure induces higher failure rates in survivors. IEEE Trans. Reliab. 37, 7374.Google Scholar
Sim, C. H. (1992). Point processes with correlated gamma interarrival times. Statist. Prob. Lett. 15, 135141.Google Scholar
Vellaisamy, P. and Sreehari, M. (2008). Some intrinsic properties of the gamma distribution. Submitted.Google Scholar
Vellaisamy, P. and Upadhye, N. S. (2007). On the negative binomial distributions and its generalizations. Statist. Prob. Lett. 77, 173180.CrossRefGoogle Scholar