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A unified approach to the study of tail probabilities of compound distributions

Part of: Special processes Distribution theory - Probability Survival analysis and censored data

Published online by Cambridge University Press:  14 July 2016

Jun Cai  and
José Garrido
Jun Cai*
Affiliation:
University of Waterloo
José Garrido*
Affiliation:
Concordia University and The University of Melbourne
*
∗Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1. Email address: jcai@setosa.uwaterloo.ca
∗∗Postal address: Department of Mathematics and Statistics, Concordia University, Montreal, Quebec, Canada H4B 1R6.
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Abstract

We consider the tail probabilities of a class of compound distributions. First, the relations between reliability distribution classes and heavy-tailed distributions are discussed. These relations reveal that many previous results on estimating the tail probabilities are not applicable to heavy-tailed distributions.

Then, a generalized Wald's identity and identities for compound geometric distributions are presented in terms of renewal processes. Using these identities, lower and upper bounds for the tail probabilities are derived in a unified way for the class of compound distributions, both under the conditions of NBU and NWU tails, which include exponential tails, as well as under the condition of heavy-tailed distributions.

Finally, simplified bounds are derived by the technique of stochastic ordering. This method removes some unnecessary technical assumptions and corrects errors in the proof of some previous results.

Keywords

Compound distributionrenewal processWald's identityreliability distribution classesNWU distributionNBU distributionheavy-tailed distributionCramér-Lundberg's conditionLundberg's inequalitystochastic ordering

MSC classification

Primary: 60E10: Characteristic functions; other transforms 60K25: Queueing theory
Secondary: 62N05: Reliability and life testing
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 4 , December 1999 , pp. 1058 - 1073
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

This research was partially funded by Montreal's Institut des Sciences Mathématiques (ISM), and the Natural Sciences and Engineering Council of Canada (NSERC) operating grant OGP0036860.

References

Alzaid, A. A. (1994). Aging concepts for items of unknown age. Comm. Statist.–Stoch. Models 10, 649–695.CrossRefGoogle Scholar
Barlow, R. E., and Proschan, F. (1981). Statistical Theory of Reliability and Life Testing: Probability Models. To Begin With, Silver Spring, MD.Google Scholar
Broeckx, F., Goovaerts, M. J., and De Vylder, F. (1986). Ordering of risks and ruin probabilities. Insur. Math. Econ. 5, 35–40.CrossRefGoogle Scholar
Cai, J., and Garrido, J. (1997). Two-sided bounds for ruin probabilities when the adjustment coefficient does not exist. Scand. Actuarial J., 80–92.Google Scholar
Cai, J., and Wu, Y. (1997). Some improvements on the Lundberg's bound for the ruin probability. Statist. Prob. Lett. 33, 395–403.CrossRefGoogle Scholar
Cao, J., and Wang, Y. (1991). The NBUC and NBUE classes of life distributions. J. Appl. Prob. 28, 473–479.CrossRefGoogle Scholar
Dickson, D. C. M. (1994). An upper bound for the probability of ultimate ruin. Scand. Actuarial J. 131–138.CrossRefGoogle Scholar
Kalashnikov, V. (1996). Two-sided bounds for ruin probabilities. Scand. Actuarial J. 1–18.CrossRefGoogle Scholar
Klefsjö, B. (1982). The HNBUE and HNWUE classes of life distributions. Naval Res. Logist. Quart. 29, 331–344.CrossRefGoogle Scholar
Lin, X. (1996). Tail of compound distributions and excess time. J. Appl. Prob. 33, 184–195.CrossRefGoogle Scholar
Panjer, H., and Willmot, G. (1992). Insurance Risk Models. Society of Actuaries, Chicago.Google Scholar
Schmidli, H. (1997). MR 97k: 60265, Mathematical Reviews, issue 97k, p. 7009 Google Scholar
Shaked, M., and Shanthikumar, J. G. (1994). Stochastic Orders and their Applications. Academic Press, New York.Google Scholar
Szekli, R. (1995). Stochastic Ordering and Dependence in Applied Probability. (Lecture Notes in Statist. 97.) Springer, New York.Google Scholar
Willmot, G. (1994). Refinements and distributional generalizations of Lundberg's inequalities. Insur. Math. Econ. 15, 49–63.CrossRefGoogle Scholar
Willmot, G. (1997a). On the relationship between bounds on the tails of compound distributions. Insur. Math. Econ. 19, 95–103.CrossRefGoogle Scholar
Willmot, G. (1997b). Bounds for compound distributions based on mean residual lifetimes and equilibrium distribution. Insur. Math. Econ. 21, 25–42.CrossRefGoogle Scholar
Willmot, G., and Lin, X. (1994). Lundberg bounds on the tails of compound distributions. J. Appl. Prob. 31, 743–756.CrossRefGoogle Scholar
Willmot, G., and Lin, X. (1997). Simplified bounds on the tails of compound distributions. J. Appl. Prob. 34, 127–133.CrossRefGoogle Scholar