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313767

Random Number Generation and Goodness -of- Fit Tests for the Lerch Distribution

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Last updated: 05 Jan 2025

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Abstract

The Lerch family of discrete distributions includes as special cases the Zipf, Zipf-Manelbrot, the logarithmic and the polylogarithmic distributions. These positively skewed long tailed distributions are used, independent from each other, in linguistics, urban economics, finance and information sciences. Zörnig and Altmann (1995) originally defined the Lerch distribution on the set of positive integers. Later, Aksenov and Savageau (2002) extend the definition to the set of nonnegative integers to allow applications in ecology, biology and genetics. In this work, we present an algorithm to compute the probabilities and moments of the Lerch distribution. This algorithm can compute the moments for areas in the parameter space that cannot be computed by Aksenov and Savageau algorithm. We derive areas on the parameter space where the variance exceeds, equal, or less than the mean. This shows the limitations on the use of the Lerch distribution to model dispersal processes, where > only for small areas in the parameter space. Then we present an efficient algorithm to generate random numbers from the Lerch distribution, and we investigate its statistical correctness property. A measure of discrepancy has been suggested and used to show that this generator has better statistical correctness properties than the Aksenov and Savageau random number generator. A modified Kolmogorov Smirnov test for goodness-of-fit for the Lerch distribution is presented and its empirical critical values have been computed.  

DOI

10.21608/esju.2004.313767

Keywords

Alias Method, EDF - Kolmogorov Smirnov and Chi-Square Goodness of-Fit Tests - Lerch Transcendent, Polylogarithm, Poisson, Binomial, geometric, Logarithmic - Power Series - Polylogarithmic and Lerch Distributions - Pseudo Random Numbers Generator - Zipf Law

Volume

48

Article Issue

2

Related Issue

42997

Issue Date

2004-12-01

Publish Date

2004-12-01

Page Start

98

Page End

113

Print ISSN

0542-1748

Online ISSN

2786-0086

Link

https://esju.journals.ekb.eg/article_313767.html

Detail API

https://esju.journals.ekb.eg/service?article_code=313767

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Original Article

Type Code

1,914

Publication Type

Journal

Publication Title

The Egyptian Statistical Journal

Publication Link

https://esju.journals.ekb.eg/

MainTitle

Random Number Generation and Goodness -of- Fit Tests for the Lerch Distribution

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Article

Created At

28 Dec 2024