Discrete Poisson Quasi-XLindley distribution with mathematical properties, regression model, and data analysis

In real-world transactions, counting data plays a crucial role. To gain a deeper understanding of this data and extract important information, statistical analysis, and modeling are necessary. This paper introduces the Poisson quasi-XLindley distribution, a novel two-parameter discrete distribution. Various mathematical characteristics of this discrete model are investigated, including mode, survival, and hazard functions, the shape of the probability mass function and failure rate (hazard function), moments, dispersion behavior, order statistics, and Shannon entropy. The parameters are estimated using the maximum likelihood approach. A simulation study is conducted to evaluate the effectiveness of the derived maximum likelihood estimators. The proposed distribution is applied to two practical datasets. Additionally, a count regression model is introduced for this distribution and applied.