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The Gompertz distribution increases exponentially and is widely used in the demographic and biological fields, especially for human mortality rates. In this blog post, we will explore the Gompertz distribution in depth and understand its applications.
Gompertz distribution basics
The Gompertz distribution was first developed by actuary Benjamin Gompertz in 1825, as a way to model age-specific mortality rates [1]. It is also called an EVD Type I distribution as it is a truncated extreme value distribution [2]. The probability density function (PDF) of the Gompertz distribution is given by [3]
Where δ and κ are shape parameters that control the shape of the probability density function.
The Gompertz distribution can also be defined by the differential equation [4]
Where r(t) is the hazard rate. When λ = 1, this equation becomes the exponential distribution.
Negative Gompertz distribution
The negative Gompertz distribution has an additional “negative rate of aging” parameter.
Gompertz supposed that the hazard rate was the probability of death at time t. He concluded that this rate increased in geometrical progression as follows:
When λ, ξ < 0, this becomes the negative Gompertz distribution.
The generalized Gompertz distribution (GGD) differs from the “regular” distribution in that
“it has increasing or constant or decreasing or bathtub curve failure rate depending upon the shape parameter…”
El-Gohary et. al [5].
Similar distributions
- The exponential distribution is composed of limits of sequences of Gompertz distributions [3].
- If X has the standard extreme value distribution for minimums, then the conditional distribution of X (given X ≥ 0 ) is the standard Gompertz distribution.
- The Exponentiated Generalized Weibull-Gompertz Distribution generalizes several distributions, including the Gompertz.
Applications
One of the unique features of the Gompertz distribution is that it is an increasing distribution with a theoretical range from zero to positive infinity. As mentioned earlier, the most common application of this distribution is in the mortality rates of human beings with a range of 0 to ≅ 100. The Gompertz model predicts that the mortality rate of human beings increases exponentially with age, which means that the older you get, the higher the probability that you will die.
Apart from human mortality rates, the Gompertz distribution finds applications in many other fields as well. For instance, it is used in biology to model the growth rate of cells in a population, where the cells multiply at an exponential rate until they reach a saturation point. Similarly, the Gompertz distribution is used in demography to model the growth rate of a population, where the birth rate decreases exponentially as the population approaches the carrying capacity.
Another interesting aspect of the Gompertz distribution is that it has a hazard function, which measures the probability of a particular event occurring at a given time, or age in our case. The hazard function of the Gompertz distribution is given by h(x) = cebx, which means that the hazard rate increases exponentially with age. This is why the Gompertz distribution is such a useful model for human mortality rates as it captures the increasing hazard rate with age accurately.
In conclusion, the Gompertz distribution, with its unique features and applications, is a fascinating probability distribution that has made a significant impact in the demographic, biological and mathematical fields. We hope that this blog post has given you a thorough understanding of the Gompertz distribution and its uses. Remember, as college students, we should always be curious and open to learning new concepts, and the Gompertz distribution is indeed a novel concept worth exploring.
References
[1] 1. Gompertz B. 1825. On the nature of the function expressive of the law of human mortality and on a new model of determining life contingencies. Phil. Trans. R. Soc. 115, 513–585. ( 10.1098/rstl.1825.0026) [PMC free article] [PubMed] [CrossRef] [Google Scholar]
[2] Johnson, N.L., Kotz, S., and Balakrishnan, N. (1994), Continuous Univariate Distributions (Vol. I, 2nd ed.), New York: Wiley.
[3] Gompertz distribution Retrieved from http://www.math.wm.edu/˜leemis/chart/UDR/UDR.html on April 12, 2023.
[4] Marshall & Olkin (2007. Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families. Springer Science & Business Media.
[5] El-Gohary, A et. al. (2013). The generalized Gompertz distribution. Applied Mathematical Modelling. Volume 37, Issues 1–2, January 2013, Pages 13–24