Approximating the Randomized Hitting Time Distribution of a Non-Stationary Gamma Process

36 Pages Posted: 1 Jun 2007

See all articles by J. B. G. Frenk

J. B. G. Frenk

Erasmus University, Rotterdam (EUR) - Department of Econometrics ; Erasmus University, Rotterdam (EUR) - Erasmus Research Institute of Management (ERIM)

Robin P. Nicolai

HKV Consultants; affiliation not provided to SSRN

Date Written: May 24, 2007

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 maintenance, First hitting time, Non-stationary gamma process, Random threshold

Suggested Citation

Frenk, J.B.G. (Hans) and Nicolai, Robin Pieter and Nicolai, Robin Pieter, Approximating the Randomized Hitting Time Distribution of a Non-Stationary Gamma Process (May 24, 2007). ERIM Report Series Reference No. ERS-2007-031-LIS, Available at SSRN: https://ssrn.com/abstract=988842

J.B.G. (Hans) Frenk (Contact Author)

Erasmus University, Rotterdam (EUR) - Department of Econometrics ( email )

P.O. Box 1738
3000 DR Rotterdam
Netherlands
+31 0 10 408 1257 (Phone)
+31 0 10 408 9162 (Fax)

Erasmus University, Rotterdam (EUR) - Erasmus Research Institute of Management (ERIM)

P.O. Box 1738
3000 DR Rotterdam
Netherlands

Robin Pieter Nicolai

HKV Consultants ( email )

Botter-11 no.29
P.O. Box 2120
Lelystad, Flevoland 8203AC
Netherlands
+31(0) 320 253901 (Fax)

HOME PAGE: http://www.hkvconsultants.com

affiliation not provided to SSRN ( email )

Botter-11 no.29
P.O. Box 2120
Lelystad, Flevoland 8203AC
Netherlands
+31(0) 320 253901 (Fax)

HOME PAGE: http://www.hkvconsultants.com

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