Transition Times: Distributions Arising from Time Heterogeneous Poisson Processes

The units of a heterogeneous population are subjected to shocks. A unit fails, or more generally, undergoes a change of state after a sufficient number of shocks. The shocks for a particular unit are assumed to arrive according to a time heterogeneous Poisson process. The time to a change of state, the transition time, for the unit has a generalized \Gamma (gamma) distribution. We assume that the intensity of the Poisson process and the number of shocks until the change of state vary independently across the units according to a \Gamma and negative binomial distribution, respectively. The distribution of the transition time is shown to be the generalized F distribution, which includes a number of standard distributions as special cases. We illustrate these results with two empirical examples: modelling coupon redemptions and traffic accidents. In the latter case, the intensity function of the Poisson process includes time varying predictor variables.