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 parameters α and β is considered. Firstly, the maximum likelihood (ML) method is used to estimate the model parameters. Asymptotic joint distributions of the estimators are obtained. Asymptotic unbiasedness and consistency statistical properties are investigated by bias and mean squared error (MSE) criteria. In the simulation study, the performance of the estimators with various values is evaluated. Finally, the applicability of the method is illustrated by using two real-life data examples. It is shown that the DGP can model the related data sets by the Kolmogorov-Smirnov (KS) test. Additionally, modified moment (MM) estimators are obtained. The (ML) estimators are compared with (MM) estimators by the (MSE) and maximum percentage error (MPE) criteria.