The double ratio geometric process for the analysis of recurrent events

Since its introduction, the geometric process (GP) has attracted extensive research attention from authors in various research communities, including probability, statistics, and reliability mathematics. However, the GP can only model a process with its gap times (i.e., times between events/failures) having a monotonic trend (either increasing or decreasing). It also implicitly assumes that the level of the modification on the hazard rate functions and that on the age after the occurrence of an event are the same, which is too restrictive and may limit its application. To overcome these drawbacks, this paper extends the GP to a new stochastic model. Probabilistic properties of the proposed model are investigated. The maximum likelihood method is used to estimate the parameters in the model. Case studies are performed to illustrate the parameter estimation process and obtain favorable performance.