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

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.