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

Science Direct Working Paper No S1574-0358(04)70255-5

30 Pages Posted: 13 Mar 2018

See all articles by Sergej V. Aksenov

Sergej V. Aksenov

affiliation not provided to SSRN

Michael A. Savageau

University of California, Davis - Department of Biomedical Engineering

Date Written: June 2002

Abstract

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.

Keywords: Discrete distribution, Lerch distribution, Lerch’s transcendent, Maximum likelihood, Method of moments, Overdispersion, Underdispersion

Suggested Citation

Aksenov, Sergej V. and Savageau, Michael A., Parameter Estimation and Random Number Generation from a Zipf-Related Lerch Distribution (June 2002). Science Direct Working Paper No S1574-0358(04)70255-5, Available at SSRN: https://ssrn.com/abstract=3137672

Sergej V. Aksenov (Contact Author)

affiliation not provided to SSRN

No Address Available

Michael A. Savageau

University of California, Davis - Department of Biomedical Engineering ( email )

United States

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