On some properties and applications of generalized weighted negative binomial distribution

In this paper, we develop a generalized class of weighted negative binomial distributions, named as weighted Gegenbauer distribution (WGbD), and study its key statistical properties. In particular, we obtain the probability generating function and derive explicit expressions for the survival function, hazard rate function, raw and factorial moments, as well as a recurrence relation for the probability mass function. We also discuss methods for estimating the parameters of the weighted Gegenbauer distribution. The proposed model is applied to real-world datasets and is shown to provide a superior fit compared to several existing models, including the Poisson distribution (PD), geometric distribution (GD), negative binomial distribution (NBD), weighted geometric distribution (WGD), generalized negative binomial distribution (GNBD), Gegenbauer distribution (GbD), and zero-inflated modified hyper-Poisson distribution (ZIMHPD). Furthermore, we examine parameter estimation in detail and develop hypothesis testing procedures to assess the significance of the model parameters. A simulation study is conducted to evaluate the performance of the maximum likelihood estimators.