Parametric and Nonparametric Estimation of Population Mean in Poisson-Xgamma Distribution with Applications to Count Data

This study proposes new estimators and confidence intervals for the population mean of the Poisson-Xgamma distribution, which are useful for overdispersed count data analysis. We prove that the proposed estimators using maximum likelihood and method of moments estimation are consistent and establish the variance of the estimators. Moreover, the confidence intervals are constructed based on large-sample theory and bootstrap method. The former method utilizes the properties of the maximum likelihood and moment estimators, the likelihood ratio, and the asymptotic normality property of the log-transformed maximum likelihood estimator. Percentile bootstrap and bias-corrected and accelerated confidence intervals are considered. The performance of the estimators is investigated through simulations in terms of bias, mean squared error, coverage probability, and length of interval. According to the simulations, the log-transformed maximum likelihood estimation-based confidence interval for the mean provides excellent and better coverage rates than the other competitive methods. Furthermore, two real data sets are used to demonstrate our estimators and perform a comparison that supports the findings obtained from the simulation study.