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SOME NEW BOUNDS FOR THE RENEWAL FUNCTION

Published online by Cambridge University Press:  06 March 2006

Konstadinos Politis  and
Markos V. Koutras
Konstadinos Politis
Affiliation:
Department of Statistics and Insurance Science, University of Piraeus, Piraeus 18534, Greece, E-mail: [email protected]; [email protected]
Markos V. Koutras
Affiliation:
Department of Statistics and Insurance Science, University of Piraeus, Piraeus 18534, Greece, E-mail: [email protected]; [email protected]
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Abstract

In the literature, most of the bounds for the renewal function U(x) corresponding to a lifetime distribution F are given in terms of the first two moments of F only. The best general upper bound of this type is the one given in Lorden (1970). In the present article, we show that improved bounds can be obtained if one exploits the specific form of the distribution F. We derive a bound that improves upon Lorden's, at least on an interval [0,a) with a ≤ ∞, and we give both sufficient and necessary conditions for this improvement to hold uniformly for x ≥ 0. Refined upper as well as lower bounds are given for the case where F belongs to a class of distributions with monotone aging or when the renewal density is monotone.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 20 , Issue 2 , April 2006 , pp. 231 - 250
Copyright
© 2006 Cambridge University Press

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References

REFERENCES

Apostol, T. (1974). Mathematical analysis. Boston: Addison Wesley.
Barlow, R.E. & Proschan, F. (1981). Statistical theory of reliability and life testing. Silver Spring, MD: To begin with.
Brown, M. (1980). Bounds, inequalities, and monotonicity properties for some specialised renewal processes. Annals of Probability 8: 227–240.Google Scholar
Brown, M. (1987). Inequalities for distributions with increasing failure rate. In A.E. Gelfand, (ed.), Contributions to the theory and applications of statistics, A volume in honor of Herbert Solomon, Orlando, FL: Academic, pp. 3–17.
Cai, J. & Garrido, J. (1999). A unified approach to the study of tail probabilities of compound distributions. Journal of Applied Probability 36(4): 1058–1073.Google Scholar
Chadjiconstantinidis, S. & Politis, K. (2005). Non-exponential bounds for stop-loss premiums and ruin probabilities. Scandinavian Actuarial Journal 5: 335–357.Google Scholar
Chadjiconstantinidis, S. & Politis, K. (2005). Two-sided bounds for the distribution of the deficit at ruin in the renewal risk model. Preprint.
Chen, Y-H. (1994). Classes of life distributions and renewal counting processes. Journal of Applied Probability 31: 1110–1115.Google Scholar
Daley, D.J. (1980). Tight bounds for the renewal function of a random walk. Annals of Probability 8: 615–621.Google Scholar
Feller, W. (1971). An introduction to probability theory and its applications, Vol. II, 2nd ed. New York: Wiley.
Lin, X.S. (1996). Tail of compound distributions and excess time. Journal of Applied Probability 33: 184–195.Google Scholar
Lorden, G. (1970). On excess over the boundary. Annals of Mathematical Statistics 41: 520–527.Google Scholar
Marshall, K.T. (1973). Linear bounds on the renewal function. SIAM Journal of Applied Mathematics 24: 245–250.Google Scholar
Politis, K. (2005). Bounds for the probability and severity of ruin in the Sparre Andersen model. Insurance: Mathematics and Economics 36(2): 165–177.Google Scholar
Ross, S. (2003). A note on the insurance risk problem. Probability in the Engineering and Informational Sciences 17: 199–203.Google Scholar
Sengupta, D. & Nanda, A.K. (1999). Log-concave and concave distributions in reliability. Naval Research Logistics 46: 419–433.Google Scholar
Shaked, M. & Zhu, H. (1992). Some results on block replacement policies and renewal theory. Journal of Applied Probability 29: 932–946.Google Scholar