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Sharp bounds for exponential approximations under a hazard rate upper bound

Part of: Markov processes Operations research and management science Special processes Distribution theory - Probability

Published online by Cambridge University Press:  30 March 2016

Mark Brown
Mark Brown*
Affiliation:
Columbia University
*
∗ Postal address: Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, NY 10027, USA. Email address: [email protected]
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Abstract

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Consider an absolutely continuous distribution on [0, ∞) with finite mean μ and hazard rate function h(t) ≤ b for all t. For bμ close to 1, we would expect F to be approximately exponential. In this paper we obtain sharp bounds for the Kolmogorov distance between F and an exponential distribution with mean μ, as well as between F and an exponential distribution with failure rate b. We apply these bounds to several examples. Applications are presented to geometric convolutions, birth and death processes, first-passage times, and to decreasing mean residual life distributions.

Keywords

Exponential approximationhazard rateKolmogorov distancegeometric convolutionbirth and death chainfirst passage timeDMRL distribution

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) 60J27: Continuous-time Markov processes on discrete state spaces 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 90B25: Reliability, availability, maintenance, inspection
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 3 , September 2015 , pp. 841 - 850
Copyright
Copyright © 2015 by the Applied Probability Trust 

References

Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York.Google Scholar
Brémaud, P. (1981). Point Processes and Queues: Martingale Dynamics. Springer, New York.Google Scholar
Brown, M. (1983). Approximating IMRL distributions by exponential distributions, with applications to first passage times. Ann. Prob. 11 419-427.Google Scholar
Brown, M. (1990). Error bounds for exponential approximations of geometric convolutions. Ann. Prob. 18 1388-1402.Google Scholar
Brown, M. (2006). Exploiting the waiting time paradox: applications of the size-biasing transformation. Prob. Eng. Inf. Sci. 20 195-230.CrossRefGoogle Scholar
Daley, D. J., Kreinin, A. Ya, and Trengove, C. D. (1992). Inequalities concerning the waiting-time in single-server queques: a survey. In Queueing and Related Models, Oxford University Press, pp. 177–223.Google Scholar
Gertsbakh, I. B. (1989). Statistical Reliability Theory. Marcel Dekker, New York.Google Scholar
Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, 2nd edn. Academic Press, New York.Google Scholar
Keilson, J. (1979). Markov Chain Models—Rarity and Exponentiality. Springer, New York.Google Scholar
Köllerström, J. (1976). Stochastic bounds for the single-server queue. Math. Proc. Cambr. Phil. Soc. 80 521-525.Google Scholar
Peköz, E. A. and Röllin, A. (2011). New rates for exponential approximation and the theorems of Rényi, and Yaglom. Ann. Prob. 39 587-608.Google Scholar
Soloviev, A. (1971). Asymptotic behaviour of the time of first occurrence of a rare event. Eng. Cybernetics 9 1038-1048.Google Scholar
Soloviev, A. D. (1972). Asymptotic distribution of the moment of first crossing of a high level by a birth and death process. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statististics and Probability, Volume III, Probability Theory, University of California Press, Berkely, pp. 71–86.Google Scholar