[Paper Review] Characterizations and Kullback-Leibler Divergence of Gompertz Distributions

Christian Bauckhage|arXiv (Cornell University)|Feb 13, 2014

Innovation Diffusion and Forecasting15 references3 citations

TL;DR

This paper characterizes the Gompertz distribution as a zero-truncated Gumbel minimum distribution and establishes its connections to extreme value distributions (Fréchet and Weibull). It derives a closed-form expression for the Kullback-Leibler divergence between two Gompertz distributions, enabling efficient model selection and statistical inference in applications such as survival analysis, income modeling, and social media attention dynamics.

ABSTRACT

In this note, we characterize the Gompertz distribution in terms of extreme value distributions and point out that it implicitly models the interplay of two antagonistic growth processes. In addition, we derive a closed form expressions for the Kullback-Leibler divergence between two Gompertz Distributions. Although the latter is rather easy to obtain, it seems not to have been widely reported before.

Motivation & Objective

• To characterize the Gompertz distribution in terms of extreme value distributions, particularly as a zero-truncated Gumbel minimum distribution.
• To interpret the Gompertz distribution as modeling the interplay of two antagonistic growth processes, providing a physical and biological rationale.
• To derive a closed-form expression for the Kullback-Leibler divergence between two Gompertz distributions, a result not widely reported in the literature.
• To support model selection and statistical inference in fields such as demography, actuarial science, marketing, and social media analysis by providing an analytically tractable divergence measure.

Proposed method

• Re-express the Gumbel distribution's probability density function (pdf) using parameters $ b $ and $ q $, showing equivalence to the Gompertz pdf after left truncation at zero.
• Demonstrate that the Gompertz distribution arises from negative logarithmic transformations of Fréchet and Weibull distributions, establishing its connection to type II and III extreme value distributions.
• Use change-of-variable techniques and the transformation rule for probability densities to derive the Gompertz pdf from Fréchet and Weibull distributions via $ y = -\ln x $ and $ y = \ln x $, respectively.
• Apply the definition of Kullback-Leibler (KL) divergence between two Gompertz densities $ f_1(x|b_1,q_1) $ and $ f_2(x|b_2,q_2) $, decomposing the integral into four components.
• Evaluate each component using substitutions $ y = e^{b_1 x} $, standard integral tables (Gradshteyn and Ryzhik), and special functions: exponential integral $ \operatorname{Ei}(-q_1) $ and upper incomplete gamma function $ \Gamma(s,x) $.
• Assemble the final closed-form expression by combining results from the four integral evaluations, yielding a complete analytical expression for the KL divergence.

Experimental results

Research questions

• RQ1 How is the Gompertz distribution related to the three types of extreme value distributions (Gumbel, Fréchet, Weibull)?
• RQ2 What is the physical or biological interpretation of the Gompertz distribution in terms of competing growth processes?
• RQ3 Can a closed-form expression for the Kullback-Leibler divergence between two Gompertz distributions be derived analytically?
• RQ4 How can this KL divergence expression be used to improve model selection or statistical inference in real-world applications?

Key findings

• The Gompertz distribution is mathematically equivalent to a zero-truncated Gumbel minimum distribution, providing a direct link to extreme value theory.
• The Gompertz distribution can be derived from both Fréchet and Weibull distributions via logarithmic or negative logarithmic transformations, confirming its role in extreme value frameworks.
• The KL divergence between two Gompertz distributions is given by a closed-form expression involving the exponential integral $ \operatorname{Ei}(-q_1) $ and the upper incomplete gamma function $ \Gamma\left(\frac{b_2}{b_1}+1, q_1\right) $, which is a novel and non-trivial result.
• The derived KL divergence expression is: $ \ln\frac{e^{q_1}b_1q_1}{e^{q_2}b_2q_2} + e^{q_1}\left[\left(\frac{b_2}{b_1}-1\right)\operatorname{Ei}(-q_1) + \frac{q_2}{q_1^{b_2/b_1}}\Gamma\left(\frac{b_2}{b_1}+1,q_1\right)\right] - (q_1 + 1) $, enabling precise statistical comparison.
• The Gompertz distribution models antagonistic growth dynamics, where the cumulative distribution function reflects a balance between accelerating and decelerating processes, such as in mortality or attention decay.
• The derived KL divergence is particularly useful for statistical inference, model selection, and comparing Gompertz models in applications like income distribution modeling, customer lifetime value prediction, and viral content dynamics.