Poisson XRani Distribution: An Alternative Discrete Distribution for Overdispersed Count Data

In this article, a one-parameter probability mass function called Poisson XRani is proposed by combining the Poisson and the XRani distributions. A wide range of distributional properties such as the shape of the probability mass function, probability generating function, factorial moments, moment generating function, characteristic function, raw moments, dispersion index, mean, variance, coefficient of skewness, and coefficient of kurtosis are investigated. It is proven that the proposed probability mass function can easily handle overdispersed and right-skewed count observations with heavy tails. Both maximum likelihood and Bayesian estimation techniques are used to estimate the unknown parameter of the proposed distribution and a simulation study was conducted to examine and analyze the performance of the maximum likelihood estimation procedure based on sample size, absolute bias, relative absolute bias, and root mean square error. The usefulness of the proposed distribution is assessed using two distinctive real datasets. These applications reveal that the new distribution provides an adequate fit compared to the other eight discrete distributions.