Some Two-Parameter Exponential Discrete Distributions and Their Applications

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In this paper we considered discrete two-parameter distribution models with variance functions in the ABM and LM classes with variance functions \(\mu(1+\frac{\mu}{p})^r\) and \(\mu(1-\frac{\mu}{p})^{-r}\) respectively (r = 1, 2, . . .). Four example frequency data sets that have received considerable attention in the literature are employed for comparative studies. We also extend our analyses to count data having co-variates (GLM cases) that exhibit over-dispersion, under-dispersion and excess zeros. Models considered here include the NB, the generalized Poisson (GP), the new logarithmic distribution (NLD), the new geometric discrete Pareto distribution (NGDP), the Poisson- Lindley-beta prime distribution (PLB) as well as the Bell-Tuchard (BTD), the discreteWeibull (DWD) and the Poisson-inverse Gaussian (PIG) distributions. All these models belong to the exponential dispersion models for count data. Several measures of comparison (-log-likelihood, Pearson’s grouped X2, Wald’s goodness-of-fit statistic \(X^2_w\) as well as the root mean square error-RMSE) were employed. Our findings indicate that the ABM and LM classes, when r ≥ 2 do not perform any better than either NB or GP with r = 1, 2 respectively and these classes are not computational flexible.