Statistical inference for doubly geometric process with Weibull interarrival times

Abstract

In recent years, the doubly geometric process is started to use as a model to fit the data from a series of events since it provides a more flexible model for wider application than the geometric process. In the applications of the doubly geometric process, the estimation problem associated with the process can arise. In this study, we deal with this problem for the stochastic process model determined by doubly geometric process when the distribution of the first interarrival time is assumed to be Weibull. The model parameters of the doubly geometric process and the parameters of Weibull distribution are estimated by using the maximum likelihood method. The joint asymptotic distribution of the estimators is obtained. We investigate some statistical properties of the estimators such as asymptotic unbiasedness and consistency. The proposed estimators are evaluated through a computer simulation study with different parameter values and sample sizes. Two goodness-of-fitness tests are applied to show that the Weibull distribution may be used for the two real world data sets which can be modeled by doubly geometric process. Finally, we present the estimators of the model and distribution parameters for these data sets.