THE POISSON-AKSHAYA DISTRIBUTION WITH PROPERTIES AND APPLICATIONS TO MODEL OVER-DISPERSED DATA

ABSTRACT The art of prediction using sample data plays a vital role in statistical modeling, particularly when dealing with count data. Among the various distributions used for modeling count data, the Poisson distribution is the most common. However, its assumption of equi-dispersion makes it unsuitable for datasets exhibiting over-dispersion, where the variance exceeds the mean. Classical models often fail in such contexts. To address this issue, we propose a new over-dispersed model based on the mixed-Poisson frame-work, termed the Poisson-Akshaya model. We derive several structural properties of the model, including factorial moments and moments about the origin. Parameter estimation is performed using the Maximum Likelihood Estimation (MLE) method and the Least Squares. A simulation study is conducted to evaluate the performance of the MLEs. Furthermore, the efficacy of the proposed model is demonstrated through its application to two real-life datasets. Based on Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and other goodness-of-fit statistics, the Poisson-Akshaya model outperforms classical and other mixed-Poisson models. Additionally, we develop the zero-inflated version of the model and assess the significance of the zero-inflation parameter using various test criteria.