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Journal Article

# Approximating IMRL Distributions by Exponential Distributions, with Applications to First Passage Times

Mark Brown
The Annals of Probability
Vol. 11, No. 2 (May, 1983), pp. 419-427
Stable URL: http://www.jstor.org/stable/2243698
Page Count: 9

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## Abstract

It is shown that if F is an IMRL (increasing mean residual life) distribution on [ 0, ∞) then: $\max\{\sup_t |\bar F(t) - \bar G(t)|, \sup_t|\bar F(t) - e^{-t/\mu}|, \sup_t|\bar G(t) - e^{-t/\mu}|, \\ \sup_t |\bar G(t) - e^{-t/\mu_G}|\} = \frac{\rho}{\rho + 1} = 1 - \frac{\mu}{\mu_G}$ where $\bar F(t) = 1 - F(t), \mu = E_FX, \mu_2 = E_FX^2, G(t) = \mu^{-1} \int^t_0 \bar F(x) dx, \mu_G = E_GX = \mu_2/2\mu$, and ρ = μ2/2μ2 - 1 = μG/μ - 1. Thus if F is IMRL and ρ is small then F and G are approximately equal and exponentially distributed. IMRL distributions with small ρ arise naturally in a class of first passage time distributions for Markov processes, as first illuminated by Keilson. The current results thus provide error bounds for exponential approximations of these distributions.

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