The geometric process has attracted extensive research attention from authors in reliability mathematics since its introduction. However, it possesses some limitations, which include that: (1) it can merely model stochastically increasing or decreasing inter-arrival times of recurrent event processes, and (2) it cannot model recurrent event processes where the inter-arrival time distributions have varying shape parameters. Those limitations may prevent it from a wider application in the real world.
In this paper, we extend the geometric process to a new process, the doubly geometric process, which overcomes the above two limitations. Probability properties are derived and two methods of parameter estimation are given. Application of the proposed model is presented: one is on fitting warranty claim data and the other is to compare the performance of the doubly geometric process with the performance of other widely used models in fitting real world datasets, based on the corrected Akaike information criterion
