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Statistical inference for geometric process with the inverse Gaussian distribution

In this study, the statistical inference problem for the geometric process (GP) is considered when the distribution of the first occurrence time is assumed to be inverse Gaussian (IG). The parameters a, μ and of the GP are estimated by using the maximum likelihood (ML) method, where a, μ and are the ratio of the GP, the mean and the variance of the IG distribution, respectively. Asymptotic distributions and consistency properties of the ML estimators are obtained. These asymptotic distributions enable us to give a test statistic which distinguishes a renewal process from a geometric process. Monte Carlo simulations are performed to compare the efficiencies of the ML estimators with the widely used nonparametric modified moment (MM) estimators. It is seen from the results that the ML estimators are more efficient than the MM estimators. Further, three real-life examples are given for application purposes.