COM-negative binomial distribution: modeling overdispersion and ultrahigh zero-inflated count data


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-inflated data sets. With the aid of ratio regression, we employ maximum likelihood method to estimate the parameters and the goodness-of-fit are evaluated by the discrete Kolmogorov-Smirnov test.

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The proposed COM-negative binomial distribution of this work was as early as conceptualized in December, 2014 when the authors saw the online version of [15]. The authors want to thank Prof. R. Köhler for mailing the valuable encyclopedia of discrete univariate distributions [39] to them. This work was partly supported by the National Natural Science Foundation of China (Grant No. 11201165).

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Zhang, H., Tan, K. & Li, B. COM-negative binomial distribution: modeling overdispersion and ultrahigh zero-inflated count data. Front. Math. China 13, 967–998 (2018).

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  • Overdispersion
  • zero-inflated data
  • infinite divisibility
  • Stein’s characterization
  • discrete Kolmogorov-Smirnov test


  • 60E07
  • 60A05
  • 62F10