Approximating the randomized hitting time distribution of a non-stationary gamma process

Hans Frenk and Robin Nicolai

No EI 2007-18, Econometric Institute Research Papers from Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute

Abstract: The non-stationary gamma process is a non-decreasing stochastic process with independent increments. By this monotonic behavior this stochastic process serves as a natural candidate for modelling time-dependent phenomena such as degradation. In condition-based maintenance the first time such a process exceeds a random threshold is used as a model for the lifetime of a device or for the random time between two successive imperfect maintenance actions. Therefore there is a need to investigate in detail the cumulative distribution function (cdf) of this so-called randomized hitting time. We first relate the cdf of the (randomized) hitting time of a non-stationary gamma process to the cdf of a related hitting time of a stationary gamma process. Even for a stationary gamma process this cdf has in general no elementary formula and its evaluation is time-consuming. Hence two approximations are proposed in this paper and both have a clear probabilistic interpretation. Numerical experiments show that these approximations are easy to evaluate and their accuracy depends on the scale parameter of the non-stationary gamma process. Finally, we also consider some special cases of randomized hitting times for which it is possible to give an elementary formula for its cdf.

Keywords: approximation; condition based maintencance; first hitting time; non-stationary gamma process; random threshold (search for similar items in EconPapers)
Date: 2007-05-10
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Citations: View citations in EconPapers (2)

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Related works:
Working Paper: Approximating the Randomized Hitting Time Distribution of a Non-stationary Gamma Process (2007)
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Persistent link: https://EconPapers.repec.org/RePEc:ems:eureir:10095

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