Poisson Sauleh Distribution and its Properties

This study introduces the Poisson-Sauleh distribution (PSuD), a novel statistical model designed to effectively handle overdispersed and heavy-tailed count data. Traditional models, such as the Poisson and Negative Binomial distributions, often fall short in accurately modeling real-world data characterized by greater variability than the mean and heavy-tailed count data. The PSuD is a mixture of the Poisson and Sauleh distributions, while Sauleh distribution is also a mixture of Exponential and Gamma distributions, to enhance flexibility and fit for complex datasets. We derive the PSuD and explore its statistical properties such as moments about the mean, variance, standard deviation, coefficient of variance, skewness, kurtosis and index of dispersion. The distribution was applied to two real-life datasets: the number of epileptic seizures and the quantity of red mites on apple leaves. Goodness-of-fit test demonstrates that the PSuD outperforms Poisson, Negative Binomial, Generalized Poisson, Poisson Janardan and Poisson Lindley Distributions, providing a more accurate representation of the underlying data characteristics. The findings indicate that the Poisson-Sauleh Distribution (PSuD) serves as a flexible and effective tool for modeling count data, particularly in cases of overdispersion and other forms of data variability across diverse fields.