The Annals of Probability

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

Mark Brown

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Abstract

It is shown that if $F$ is an IMRL (increasing mean residual life) distribution on $\lbrack 0, \infty)$ 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 $\rho = \mu_2/2\mu^2 - 1 = \mu_G/\mu - 1$. Thus if $F$ is IMRL and $\rho$ is small then $F$ and $G$ are approximately equal and exponentially distributed. IMRL distributions with small $\rho$ 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.

Article information

Source
Ann. Probab., Volume 11, Number 2 (1983), 419-427.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993607

Digital Object Identifier
doi:10.1214/aop/1176993607

Mathematical Reviews number (MathSciNet)
MR690139

Zentralblatt MATH identifier
0519.62015

JSTOR
links.jstor.org

Subjects
Primary: 62E10: Characterization and structure theory
Secondary: 60K10: Applications (reliability, demand theory, etc.)

Keywords
IMRL distributions approximate exponentiality first passage times Markov processes reliability theory inequalities

Citation

Brown, Mark. Approximating IMRL Distributions by Exponential Distributions, with Applications to First Passage Times. Ann. Probab. 11 (1983), no. 2, 419--427. doi:10.1214/aop/1176993607. https://projecteuclid.org/euclid.aop/1176993607


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Corrections

  • See Correction: Mark Brown. Corrections: Correction to "Approximating IMRL Distributions by Exponential Distributions, with Applications to First Passage Times. Ann. Probab., Volume 11, Number 4 (1983), 1055--1055.