[Paper Review] Some properties of the Lerch family of discrete distributions

Sergey Aksenov, Michael A. Savageau|ArXiv.org|Apr 23, 2005

Bayesian Methods and Mixture Models15 references9 citations

TL;DR

This paper extends the Lerch distribution to nonnegative integers, enabling modeling of count data with zero counts. It derives closed-form expressions for key properties—mean, variance, hazard function, and variance-to-mean ratio—using Lerch's transcendent, demonstrating the distribution's flexibility for underdispersed and overdispersed data, with superior fit to sea urchin sperm fertilization data compared to generalized Poisson models.

ABSTRACT

We extend the definition of the Lerch distribution to the set of nonnegative integers for greater applicability to modeling count data. We express its properties in terms of Lerch's transcendent, and study its unimodality, hazard function and variance-to-mean ratio.

Motivation & Objective

• To extend the Lerch distribution from positive integers to nonnegative integers for broader applicability in count data modeling.
• To express all distributional properties—mean, variance, hazard function, and variance-to-mean ratio—explicitly in terms of Lerch’s transcendent.
• To demonstrate the distribution’s flexibility in modeling both underdispersed and overdispersed data, especially where standard Poisson or generalized Poisson models fail.
• To provide a practical computational framework via a Mathematica package for parameter fitting and statistical inference.

Proposed method

• Define the probability mass function as $ p_x = \frac{c z^x}{(v + x)^s} $, where $ c = 1 / \Phi(z, s, v) $, and $ \Phi $ is Lerch’s transcendent.
• Use the functional equation of Lerch’s transcendent to derive the cumulative distribution function: $ F(x) = 1 - z^{x+1} \frac{\Phi(z, s, v + x + 1)}{\Phi(z, s, v)} $.
• Derive the mean and variance via series expansion in powers of $ z $, up to second order, to analyze the variance-to-mean ratio.
• Employ the Pearson $ X^2 $ statistic minimization for parameter estimation, especially effective for sparse data with few frequency classes.
• Implement a custom Mathematica package, LerchDistribution.m, to extend standard statistical functions to include the Lerch distribution and enable parameter fitting.
• Leverage a convergence acceleration technique for efficient computation of Lerch’s transcendent, available in C and Mathematica code.

Experimental results

Research questions

• RQ1 Can the Lerch distribution be meaningfully extended from positive integers to nonnegative integers to better model count data with zero counts?
• RQ2 How do the hazard function and variance-to-mean ratio of the extended Lerch distribution behave across different parameter values, and can they model both underdispersion and overdispersion?
• RQ3 Does the extended Lerch distribution provide a better fit than the generalized Poisson distribution for underdispersed count data, such as sea urchin sperm fertilization counts?
• RQ4 Can the variance-to-mean ratio be analytically expressed in terms of the parameters $ z $, $ s $, and $ v $, and what conditions on $ s $ determine whether the distribution is under-, equi-, or overdispersed?
• RQ5 Is the Pearson $ X^2 $ statistic minimization a robust and practical method for parameter estimation in the Lerch distribution, especially with limited data?

Key findings

• The extended Lerch distribution provides a better fit than the generalized Poisson model for underdispersed sea urchin sperm fertilization data, with sum of squared deviations of 0.000372774 (40 sec) and 0.000162184 (180 sec).
• The variance-to-mean ratio can be less than, equal to, or greater than one depending on the parameter $ s $, with a critical threshold at $ s = -\log 2 / \log(1 + 1/(v^2 + 2v)) $, enabling modeling of both underdispersed and overdispersed data.
• For small $ z $, the variance-to-mean ratio is less than one when $ s < -\log 2 / \log(1 + 1/(v^2 + 2v)) $, and greater than one when $ s $ exceeds this threshold.
• The best-fit parameters for the 40-sec data are $ z = 0.00773867 $, $ s = -8.26894 $, $ v = 1.11633 $, and for 180-sec data, $ z = 0.0835808 $, $ s = -1.15174 $, $ v = 0.00468234 $, indicating strong model fit.
• The method of minimizing the Pearson $ X^2 $ statistic is shown to be more practical and accurate than moment or maximum likelihood methods for small data sets with sparse frequency classes.
• The distribution’s hazard function can be constant, increasing, or decreasing depending on parameters, making it suitable for modeling diverse survival and dispersal processes.