Approximate confidence and tolerance limits for the discrete Pareto distribution for characterizing extremes in count data

Statistical tolerance intervals for discrete distributions are widely employed for assessing the magnitude of discrete characteristics of interest in applications like quality control, environmental monitoring, and the validation of medical devices. For such data problems, characterizing extreme counts or outliers is also of considerable interest. These applications typically use traditional discrete distributions, like the Poisson, binomial, and negative binomial. The discrete Pareto distribution is an alternative yet flexible model for count data that are heavily right-skewed. Our contribution is the development of statistical tolerance limits for the discrete Pareto distribution as a strategy for characterizing the extremeness of observed counts in the tail. We discuss the coverage probabilities of our procedure in the broader context of known coverage issues for statistical intervals for discrete distributions. We address this issue by applying a bootstrap calibration to the confidence level of the asymptotic confidence interval for the discrete Pareto distribution's parameter. We illustrate our procedure on a dataset involving cyst formation in mice kidneys.