COM-negative binomial distribution: modeling overdispersion and ultrahigh zero-inated count data
Front. Math. China. 2018, 13 (4): 967-998.
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We focus on the COM-type negative binomial distribution with three parameters, which belongs to COM-type (a, b, 0) class distributions and family of equilibrium distributions of arbitrary birth-death process. Besides, we show abundant distributional properties such as overdispersion and underdispersion, log-concavity, log-convexity (infinite divisibility), pseudo compound Poisson, stochastic ordering, and asymptotic approximation. Some characterizations including sum of equicorrelated geometrically distributed random variables, conditional distribution, limit distribution of COM-negative hypergeometric distribution, and Stein's identity are given for theoretical properties. COM-negative binomial distribution was applied to overdispersion and ultrahigh zero-inated data sets. With the aid of ratio regression, we employ maximum likeli-hood method to estimate the parameters and the goodness-of-fit are evaluated by the discrete Kolmogorov-Smirnov test.