Bivariate geometric distribution
Probability generating function
Empirical probability generating function
We propose two types of goodness-of-fit tests for geometric distribution and for a bivariate geometric distribution called BGD(B&D), based on their probability generating function (PGF). The first type is a special-case application of the general testing procedure for discrete distributions proposed by Kocherlakota and Kocherlakota (1986). The second type utilizes the supremum of the absolute value of the standardized difference between the PGF’s maximum likelihood estimator (MLE) and its empirical counterpart as the test statistic. We verify the asymptotic properties of the test statistics for the first type of test and explore the asymptotic behaviors of the test statistics for the second type of test by calculating the empirical critical points and constructing the density curves. We compare the proposed tests with Chi-square and the empirical distribution function (EDF) related tests proposed in the literature in terms of significance level and power. Based on the comparison results, we recommend the second type of goodness-of fit test for both geometric distribution and BGD(B&D) because of its robustness, efficiency in computation and no need for selecting t. Real data sets are used for illustration.