A New Discrete Analogue of the Continuous Muth Distribution for Over-Dispersed Data: Properties, Estimation Techniques, and Application.

We present a new one-parameter discrete Muth (DsMuth) distribution, a flexible probability mass function designed for modeling count data, particularly over-dispersed data. The proposed distribution is derived through the survival discretization approach. Several of the proposed distribution's characteristics and reliability measures are investigated, including the mean, variance, skewness, kurtosis, probability-generating function, moments, moment-generating function, mean residual life, quantile function, and entropy. Different estimation approaches, including maximum likelihood, moments, and proportion, are explored to identify unknown distribution parameters. The performance of these estimators is assessed through simulations under different parameter settings and sample sizes. Additionally, a real dataset is used to emphasize the significance of the proposed distribution compared to other available discrete probability distributions.