The Pareto–Poisson Distribution: Characteristics, Estimations and Engineering Applications

A new three-parameter lifetime distribution based on compounding Pareto and Poisson distributions is introduced and discussed. Various statistical and reliability properties of the proposed distribution including: quantiles, ordinary moments, median, mode, quartiles, mean deviations, cumulants, generating functions, entropies, mean residual life, order statistics and stress-strength reliability are obtained. In presence of data collected under Type-II censoring, from frequentist and Bayesian points of view, the model parameters are estimated. Using independent gamma priors, Bayes estimators against the squared-error, linear-exponential and general-entropy loss functions are developed. Based on asymptotic properties of the classical estimators, asymptotic confidence intervals of the unknown parameters are constructed using observed Fisher’s information. Since the Bayes estimators cannot be obtained in closed-form, Markov chain Monte Carlo techniques are considered to approximate the Bayes estimates and to construct the highest posterior density intervals. A Monte Carlo simulation study is conducted to examine the performance of the proposed methods using various choices of effective sample size. To highlight the perspectives of the utility and flexibility of the new distribution, two numerical applications using real engineering data sets are investigated and showed that the proposed model fits well compared to other eleven lifetime models.