Parameter Estimation and Random Number Generation from a Zipf-Related Lerch Distribution

The Lerch family of three-parameter, discrete univariate distributions includes as special cases the well known Zipf, Zipf-Mandelbrot, and Good distributions that are used as models in ecology, linguistics, information science, and statistical physics. The Lerch distribution was originally defined on the set of positive integers by Zörnig and Altmann (1995, Comp. Stat. Data An., 19, 461). Here we extend the definition of the Lerch distribution to the set of nonnegative integers, which allows application of the Lerch distribution to problems where the observed random variable can take zero values, e.g. modeling count data and survival and dispersal processes in ecology. Properties of truncated distributions, including the zero-truncated case studied by Zörnig and Altmann, can be easily derived from the general Lerch distribution. We derive conditions on parameters of the general Lerch distribution for it to be strongly unimodal, give formulas for its uncorrected, central, and factorial moments, and derive the moment and maximum likelihood estimators. We also propose an algorithm for random number generation from the Lerch distribution that is based on the method of inversion by correction. Finally, we provide a Mathematica package for calculations related to the Lerch distribution