Discrete Poisson Haq distribution with mathematical properties and count data modeling.

Area of Science:

• Statistics
• Probability Theory
• Biostatistics

Background:

• Count data often exhibit over-dispersion, violating assumptions of standard Poisson distributions.
• Existing discrete distributions may not adequately capture the complexities of over-dispersed medical count data.
• Need for flexible statistical models to analyze various biological and medical count datasets.

Purpose of the Study:

• To propose a new one-parameter discrete distribution, the Poisson Haq (PH) distribution.
• To analyze over-dispersed count datasets, particularly in medical contexts.
• To develop and evaluate a parametric regression model based on the PH distribution.

Main Methods:

• The Poisson Haq distribution is constructed as a mixture of Poisson and Haq random variables.
• Statistical properties, including failure rate shapes (increasing and upside bathtub), are derived.
• Parameter estimation is performed using the method of moments, maximum likelihood estimation, and Bayesian approaches with a gamma prior.

Main Results:

• The PH distribution effectively models over-dispersed count data, showing improved fitting accuracy (lower AIC and BIC values) compared to Poisson, Poisson moment exponential, and Poisson-XLindley distributions.
• Simulation studies confirm the performance and behavior of the proposed estimators.
• The PH regression model demonstrates good fit for the Length of Hospital Stay dataset.

Conclusions:

• The proposed Poisson Haq distribution offers a valuable and flexible alternative for analyzing over-dispersed count data in medical and biological research.
• The PH distribution and its associated regression model provide enhanced accuracy and better data-fitting capabilities.
• The model's applicability is validated across diverse medical datasets, including infectious diseases and cytogenetic lesions.