Characterizations and Kullback-Leibler Divergence of Gompertz Distributions

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. 1 The Gompertz Distribution The Gompertz distribution provides a statistical formulation of the Gompertz law of mortality [1]. Its probability density function (pdf) is defined for x ∈ [0,∞) and given by f(x | b, q) = e b q e e−q e bx (1) where the parameter b > 0 determines scale and q > 0 is a shape parameter. The corresponding cumulative density function (cdf) amounts to F (x | b, q) = 1− e e−q e bx (2) and will be of interest in our discussion below. Regarding the density in (1), we note that it is unimodal and rather flexible. Depending on the choice of b and q, it may be skewed to the left or to the right; however, for q ≥ 1, its mode will always be at 0. Due to its origins as a model of mortality, the Gompertz distribution is a staple in statistical biology and the demographic and actuarial sciences [2,3]. It was observed to model income distributions [4] and has been used as a model of the diffusion of novel products as well as of customer life-time values [5,6,7] in economics and marketing . Finally, in the context of social media analysis, the Gompertz distribution was found to account well for the temporal evolution of collective attention to viral Web content or social media services [8,9]. Our goal with this note is to provide an accessible account of some of the properties of the Gompertz distribution. Furthermore, we derive a closed form expression for the Kullback-Leibler divergence between Gompertz distributions which is useful for the purpose of model selection or statistical inference. 2 Interpretation in Terms of Extreme Value Distributions Interestingly, the Gompertz distribution is rather closely related to extreme value theory. Here, we briefly demonstrate that it can be expressed in terms of the three extreme value distributions. First of all, the Gompertz distribution corresponds to a zero-truncated Gumbel minimum distribution. The Gumbel distribution is the type I extreme value distribution. When used to model the distribution of sample minima, its pdf is defined for x ∈ (−∞,∞) and usually expressed as fG(x | m, s) = 1 s e(x−m)/s e−e (x−m)/s = 1 s e x s e− m s e−e x s e− m s (3) where m is a location parameter and s > 0 determines scale. Hence, defining b = 1s and q = e −m/s allows us to re-parameterize (3) and to write it as fG(x | b, q) = b q e e−q e bx (4) such that the corresponding cumulative density function amounts to FG(x | b, q) = 1− e−q e bx . (5) Looking at the cumulative density in (5), we note that limx→∞ FG(x) = 1 as well as FG(0) = 1− e−q. Accordingly, by left truncating the density in (4) at 0, we obtain a distribution whose pdf is given by fG(x) ∫∞ 0 fG(x)dx = fG(x) 1− FG(0) = fG(x) 1− (1− e−q) = efG(x) = e b q e e−q e bx . (6) This, however, is indeed the probability density of the Gompertz distribution as introduced in (1). Second of all, the Gompertz is indirectly related to the Fréchet and to the Weibull distribution. The Fréchet distribution is the type II extreme value distribution. It is usually defined for x ∈ (0,∞) in which case its pdf is given by fF (x | a, r) = a r (x r )−1−a e−( x r ) −a (7) where a > 0 and r > 0 are shape and scale parameters, respectively. The Weibull distribution is the type III extreme value distribution. It is commonly defined for x ∈ [0,∞) and its pdf amount to fW(x | k, l) = k l (x l )k−1 e−( x l ) k (8) where k > 0 and l > 0 are shape and scale parameters, respectively. In order to expose the connections between the densities in (7) and (8) and the Gompertz density in (1), we recall that if a random variable X is distributed according to fX(x), the monotonously transformed random variable Y = h(X) has a pdf that is given by fY (y) = fX ( h−1(y) ) ∣∣∣∣ d dyh−1(y) ∣∣∣∣ . (9) Using this identity, it is straightforward to see that the Gompertz distribution also results from transforming Fréchet or Weibull distributions. In particular, if fX(x) is a Fréchet density and y = − lnx, then x = e−y and dx dy = −e−y (10)