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On higher-order properties of compound geometric distributions

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Gordon E. Willmot
Gordon E. Willmot*
Affiliation:
University of Waterloo
*
∗ Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1. Email address: [email protected]
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Abstract

An explicit convolution representation for the equilibrium residual lifetime distribution of compound zero-modified geometric distributions is derived. Second-order reliability properties are seen to be essentially preserved under geometric compounding, and complement results of Brown (1990) and Cai and Kalashnikov (2000). The convolution representation is then extended to the nth-order equilibrium distribution. This higher-order convolution representation is used to evaluate the stop-loss premium and higher stop-loss moments of the compound zero-modified geometric distribution, as well as the Laplace transform of the kth moment of the time of ruin in the classical risk model.

Keywords

Compound geometric convolutionresidual lifetime distributionequilibrium distributionintegrated taildefective renewal equationDFRIFRIMRLDMRLNWUNBU2-NWU2-NBUNWUCNBUCNWUENBUEmean residual lifetimeLaplace-Stieltjes transformstop-loss premiumstop-loss momentsLundberg boundtime of ruin

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 2 , June 2002 , pp. 324 - 340
Copyright
Copyright © Applied Probability Trust 2002 

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References

Asmussen, S. (2000). Ruin Probabilities. World Scientific, Singapore.CrossRefGoogle Scholar
Barlow, R., and Proschan, F. (1965). Mathematical Theory of Reliability. John Wiley, New York.Google Scholar
Brown, M. (1990). Error bounds for exponential approximations of geometric convolutions. Ann. Prob. 18, 1388–1402.CrossRefGoogle Scholar
Cai, J., and Kalashnikov, V. (2000). NWU property of a class of random sums. J. Appl. Prob. 37, 283–289.CrossRefGoogle Scholar
Delbaen, F. (1990). A remark on the moments of ruin time in classical risk theory. Insurance Math. Econom. 9, 121–126.CrossRefGoogle Scholar
Esary, J., Marshall, A., and Proschan, F. (1973). Shock models and wear processes. Ann. Prob. 1, 627–649.CrossRefGoogle Scholar
Fagiuoli, E., and Pellerey, F. (1993). New partial orderings and applications. Naval Res. Logistics 40, 829–842.3.0.CO;2-D>CrossRefGoogle Scholar
Fagiuoli, E., and Pellerey, F. (1994). Preservation of certain classes of life distributions under Poisson shock models. J. Appl. Prob. 31, 458–465.CrossRefGoogle Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Gertsbakh, I. (1984). Asymptotic methods in reliability theory: a review. Adv. Appl. Prob. 16, 147–175.CrossRefGoogle Scholar
Goovaerts, M., De Vylder, F., and Haezendonck, J. (1984). Insurance Premiums. North Holland, Amsterdam.Google Scholar
Hesselager, O., Wang, S., and Willmot, G. (1997). Exponential and scale mixtures and equilibrium distributions. Scand. Actuarial J. 1997, 125–142.Google Scholar
Kaas, R., van Heerwaarden, A., and Goovaerts, M. (1994). Ordering of Actuarial Risks. CAIRE, Brussels.Google Scholar
Kaas, R., Goovaerts, M., Dhaene, J., and Denuit, M. (2001). Modern Actuarial Risk Theory. Kluwer, Dordrecht.Google Scholar
Kalashnikov, V. (1997). Geometric Sums: Bounds for Rare Events with Applications. Kluwer, Dordrecht.CrossRefGoogle Scholar
Klugman, S., Panjer, H., and Willmot, G. E. (1998). Loss Models—From Data to Decisions. John Wiley, New York.Google Scholar
Lin, X., and Willmot, G. E. (2000). The moments of the time of ruin, the surplus before ruin, and the deficit at ruin. Insurance Math. Econom. 27, 19–44.CrossRefGoogle Scholar
Nanda, A. K., Jain, H., and Singh, H. (1996). On closure of some partial orderings under mixtures. J. Appl. Prob. 33, 698–706.CrossRefGoogle Scholar
Neuts, M. (1986). Generalizations of the Pollaczek–Khinchin integral equation in the theory of queues. Adv. Appl. Prob. 18, 952–990.CrossRefGoogle Scholar
Rolski, T., Schmidli, H., Schmidt, V., and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.CrossRefGoogle Scholar
Shaked, M., and Shanthikumar, J. (1994). Stochastic Orders and Their Applications. Academic Press, San Diego.Google Scholar
Shanthikumar, J. (1988). DFR property of first passage times and its preservation under geometric compounding. Ann. Prob. 16, 397–406.CrossRefGoogle Scholar
Sundt, B. (1982). Asymptotic behaviour of compound distributions and stop-loss premiums. Astin Bull. 13, 89–98. (Corrigendum 15 (1985), 44.)CrossRefGoogle Scholar
Szekli, R. (1995). Stochastic Ordering and Dependence in Applied Probability. Springer, New York.CrossRefGoogle Scholar
Willmot, G. E. (2002). Compound geometric residual lifetime distributions and the deficit at ruin. Submitted. To appear in Insurance Math. Econom.CrossRefGoogle Scholar
Willmot, G. E., and Lin, X. (2001). Lundberg Approximations for Compound Distributions with Insurance Applications. Springer, New York.CrossRefGoogle Scholar
Willmot, G. E., Cai, J., and Lin, X. (2001). Lundberg inequalities for renewal equations. Adv. Appl. Prob. 33, 674–689.CrossRefGoogle Scholar