Approximate tolerance limits for Zipf–Mandelbrot distributions

Zipf–Mandelbrot distributions are commonly used to model natural phenomena where the frequency of an event’s occurrence is inversely proportional to its rank based on that frequency of occurrence. This discrete distribution typically exhibits a large number of rare events; however, it may be of interest to obtain reasonable limits that bound the majority of the number of different events. We propose the use of statistical tolerance limits as a way to quantify such a bound. The tolerance limits are constructed using Wald confidence limits for the Zipf–Mandelbrot parameters and are shown through a simulation study to have coverage probabilities near the nominal levels. We also calculate Zipf–Mandelbrot tolerance limits for two real datasets and discuss the associated computer code developed for the R programming language.