Parameter Estimation for Doubly Geometric Process with Inverse Gaussian Distribution and its Applications

The geometric process (GP) has an important role in the reliability theory. Although it is used as a stochastic model in various areas of application, it has some limitations to fit the data purposes of various empirical studies. The doubly geometric process (DGP) has been proposed to overcome these limitations. The parameter estimation problem is very important 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 inverse Gaussian (IG) distribution with parameters [Formula: see text], [Formula: see text] is considered. First, the model parameters are estimated by using the maximum likelihood (ML) method, and then for obtained estimators, asymptotic joint distribution and asymptotic unbiasedness and consistency properties are investigated. A test statistic is introduced based on ML methods to distinguish DGP from GP. The non-parametric (NP) methods which are least square (LSE) and log-LSE are presented. Additionally, modified moment (MM) estimators are obtained. The efficiencies of the ML estimators are compared with MM estimators by an extensive simulation study. It is concluded that the ML estimators perform better than the MM estimators. Finally, three real-life data examples are presented to illustrate the applicability of the method. It is shown that the DGP can model the related data sets.