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.

Approximation, Condition based maintenance, First hitting time, Non-stationary gamma process, Random threshold
Forecasting and Other Model Applications (jel C53), Business Administration and Business Economics; Marketing; Accounting (jel M), Production Management (jel M11), Innovation and Invention: Processes and Incentives (jel O31), Management of Technological Innovation and R&D (jel O32), Technological Change: Choices and Consequences; Diffusion Processes (jel O33)
Erasmus Research Institute of Management
hdl.handle.net/1765/10149
ERIM Report Series Research in Management
ERIM report series research in management Erasmus Research Institute of Management
Erasmus Research Institute of Management

Frenk, J.B.G, & Nicolai, R.P. (2007). Approximating the Randomized Hitting Time Distribution of a Non-stationary Gamma Process (No. ERS-2007-031-LIS). ERIM report series research in management Erasmus Research Institute of Management. Erasmus Research Institute of Management. Retrieved from http://hdl.handle.net/1765/10149