On the application of Ferguson characterization for the construction of a goodness-of-fit test for the geometric distribution

We propose two novel goodness-of-fit tests tailored for the geometric distribution. In contrast to the commonly used approach to base statistics on the empirical counterpart of the characterizing differential equation whose solution is the probability generating function of the distribution of interest, we base our tests on an independence-type characterization of the geometric distribution. The proposed test statistics rely on the discrepancy between the joint and the product of marginal V-empirical probability generating functions that correspond to the functions appearing in the characterization. We derive the asymptotic null distributions of the test statistics and their almost sure limits under general conditions. To evaluate the quality of the introduced tests, we conduct a comprehensive empirical power study to assess their finite sample properties. Our findings indicate that the proposed tests demonstrate competitive performance. Furthermore, we demonstrate the practical applicability of these tests by applying them to various real datasets.