Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-14T16:56:58.037Z Has data issue: false hasContentIssue false
  • English
  • Français

Tail of compound distributions and excess time

Part of:Special processes

Published online by Cambridge University Press:  14 July 2016

Xiaodong Lin
Xiaodong Lin*
Affiliation:
University of Toronto
*
∗ Postal address: Department of Statistics, University of Toronto, Toronto, Ontario M5S 1A1, Canada.
Get accessRights & Permissions [Opens in a new window]

Abstract

Bounds on the tail of compound distributions are considered. Using a generalization of Wald's fundamental identity, we derive upper and lower bounds for various compound distributions in terms of new worse than used (NWU) and new better than used (NBU) distributions respectively. Simple bounds are obtained when the claim size distribution is NWUC, NBUC, NWU, NBU, IMRL, DMRL, DFR and IFR. Examples on how to use these bounds are given.

Keywords

EXCESS TIMEEXPONENTIAL AND NON-EXPONENTIAL BOUNDSINSURANCE RISKNEW BETTER THAN USEDNEW WORSE THAN USEDRUIN PROBABILITYWALD'S IDENTITYM/G/1 QUEUE

MSC classification

Primary:60K25: Queueing theory

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 1 , March 1996, pp. 184 - 195
Copyright
Copyright © Applied Probability Trust 1996 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Asmussen, S. (1987) Applied Probability and Queues. Wiley, Chichester.Google Scholar
Asmussen, S. (1994) Personal communication.Google Scholar
Barlow, R. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York.Google Scholar
Bowers, N., Gerber, H., Hickman, J., Jones, D. and Nesbitt, C. (1986) Actuarial Mathematics. Society of Actuaries, Ithaca.Google Scholar
Cao, J. and Wang, Y. (1991) The NBUC and NWUC classes of life distributions. J. Appl. Prob. 28, 473–479; Correction (1992) 29, 753.Google Scholar
Embrechts, P. (1983) A property of the inverse Guassian distribution with some applications. J. Appl. Prob. 20, 537–544.Google Scholar
Embrechts, P. and Goldie, C. (1982) On convolution tails. Stoch. Proc. Appl. 13, 263–278.CrossRefGoogle Scholar
Gerber, H. (1973) Martingales in risk theory. Verein. Schweiz. Versiche. Math. Mitteil. 73, 205–216.Google Scholar
Gerber, H. (1979) An Introduction to Mathematical Risk Theory. Huebner Foundation, University of Pennsylvania, PA.Google Scholar
Gerber, H. (1994) Martingales and tail probabilities. Austin Bulletin 24, 145–146.CrossRefGoogle Scholar
Gertsbakh, I. (1989) Statistical Reliability Theory. Marcel Dekker, New York.Google Scholar
Kalashnikov, V. (1993) Two-side estimates of geometric convolutions. In Stability Problems for Stochastic Models. (Lecture Notes in Math. 1546), pp. 75–88. ed. Kalashnikov, V. Springer, Berlin.Google Scholar
Kingman, J. F. C. (1964) A martingale inequality in the theory of queues. Proc. Camb. Phil. Soc. 60, 359–361.Google Scholar
Kingman, J. F. C. (1970) Inequalities in the theory of queues. J. R. Stat. Soc. B32, 102–110.Google Scholar
Panjer, H. and Willmot, G. (1992) Insurance Risk Models. Society of Actuaries, Ithaca.Google Scholar
Ross, S. (1983) Stochastic Processes. Wiley, New York.Google Scholar
Ross, S. (1974) Bounds on the delay distribution in GI/G/1 queues. J. Appl. Prob. 11, 417–421.Google Scholar
Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. ed. Daley, D. Wiley, Chichester.Google Scholar
Taylor, G. C. (1976) Use of differential and integral inequalities to bound ruin and queueing probabilities. Scand. Act. J. 197–208.Google Scholar
Wald, A. (1944) On cumulative sums of random variables. Ann. Math. Statist. 15, 283–296.Google Scholar
Wald, A. (1946) Differentiation under the expectation sign in the fundamental identity of sequential analysis. Ann. Math. Statist. 17, 493–497.Google Scholar
Willmot, G. (1994) Refinements and distributional generalizations of Lundberg's inequalities. Insurance: Mathematics and Economics 15, 49–63.Google Scholar
Willmot, G. and Lin, X. (1994). Lundberg bounds on the tails of compound distributions. J. Appl. Prob. 31, 743–756.CrossRefGoogle Scholar