A new extension of Poisson distribution for asymmetric count data: theory, classical and Bayesian estimation with application to lifetime data

Several research investigations have stressed the importance of discrete data analysis and its relevance to actual events. The current work focuses on a new discrete distribution with a single parameter that can be derived using the Poisson mixing technique. The new distribution is named the Poisson Entropy-Based Weighted Exponential Distribution. It is useful for discussing asymmetric "right-skewed" data with "heavy" tails. Its failure rate function can be used to explain situations with increasing failure rates. The statistical properties of the new distribution are expressed explicitly. The proposed model is simple to manage for under-, equal-, and over-dispersed datasets. The model parameters are estimated using the maximum likelihood method. We consider the parameter estimation for the new model based on right-censored data with a cure fraction. One more focus of the present study is the Bayesian estimation of the model parameters. In the end, three real-world dataset examples were utilized to show the value of the new distribution. These applications revealed that the new model outperforms other standard discrete models.