  In this paper, 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.
