In this article, we propose a general family of distributions called odd Burr III G-negative binomial. Several properties of the family are reported, including quantile function, mixture representation of its density, cumulative distribution functions, moments, incomplete moments, moment generating function, mean deviations, deviations, stochastic ordering, entropies, and parameter estimation via maximum likelihood (ML) method. A simulation study is carried out to check the asymptotic behavior of the ML estimates. The flexibility of the governing members of this family is shown by the use of data modeling such that the corresponding density functions may be symmetric, right-skewed, or left-skewed and their hazard rate functions can be increasing, decreasing, bathtub, upside-down bathtub, or constant.
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