Parameter Estimation Problem for Doubly Geometric Process with the Gamma Distribution and Some Applications

The geometric process (GP) is one of the important and widely used stochastic models in reliability theory. Although it is used in various areas of application, it has some limitations that cause difficulties. The doubly geometric (DGP) has been proposed to overcome these limitations. The parameter estimation problem plays an important role for both GP and DGP. In this study, the parameter estimation problem for DGP when the distribution of the first interarrival time is assumed to be a gamma distribution with the parameters α and β is considered. Firstly, the maximum likelihood (ML) method is used to estimate the model parameters. Asymptotic joint distribution of the estimators and their asymptotic unbiasedness and consistency properties are obtained. Then the small sample performances of the estimators are evaluated by a simulation study. Finally, the applicability of the method is illustrated by using two real-life data examples. It is shown that these data sets can be modeled by DGP. Additionally, the nonparametric estimators which are called modified moment (MM) estimators are compared with the ML estimators. As a result it can be said that the ML estimators are more efficient than the MM estimators.