Random Number Generation and Goodness -of- Fit Tests for the Lerch Distribution | ||
| The Egyptian Statistical Journal | ||
| Article 1, Volume 48, Issue 2, December 2004, Pages 98-113 | ||
| Document Type: Original Article | ||
| DOI: 10.21608/esju.2004.313767 | ||
| 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. | ||
| 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 | ||
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