Discrete Poisson Haq distribution with mathematical properties and count data modeling

A new one-parameter discrete distribution, namely the Poisson Haq (PH) distribution, is proposed by a mixture of the Poisson variable and an independently distributed Haq random variable. This model effectively analyzes over-dispersed count datasets by extending Poisson distribution. Various useful statistical properties of the PH distribution are derived and discussed. The failure rate of the proposed distribution is "increasing" and "upside bathtub" shaped. The model parameter estimation is performed using renowned estimation approaches, method of moments, and method of maximum likelihood estimation. A parametric regression model tailored for count datasets is also developed using the proposed distribution. A simulation study is conducted to demonstrate the performance and behavior of the proposed estimators. The present study validates that the new count model adequately explains the medical datasets, which are the number of infected patients with the Nipah virus, the number of mammalian cytogenetic dosimetry lesions, and the Length of Hospital Stay. Additionally, we also estimate the model parameter using the Bayesian approach with gamma prior. Compared to widely used alternatives such as the Poisson (AIC = 145.16, BIC = 147.19), Poisson moment exponential (AIC = 137.53, BIC = 139.56), Poisson-XLindley (AIC = 135.86, BIC = 137.88) distributions and others, our model demonstrates improved fitting accuracy, as evidenced by lower AIC (135.78) and BIC (137.81) values for first data and similarly for second data applications. Finally, to validate the fit of the PH regression model, it is applied to the Length of Hospital Stay dataset.