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Sharp bounds for exponential approximations under a hazard rate upper bound

Part of: Special processes Operations research and management science Markov 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: mb2484@columbia.edu
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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 

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