Calculate the Gini index for the Gompertz distribution with scale parameter \(\beta\) and shape parameter \(\alpha\).
ggompertz(
scale = 1,
shape
)A numeric value with the Gini index. A NA is returned when a parameter is non-numeric or non-positive.
A positive real number specifying the scale parameter \(\beta\) of the Gompertz distribution. The default value is scale = 1.
A positive real number specifying the shape parameter \(\alpha\) of the Gompertz distribution.
Juan F Munoz jfmunoz@ugr.es
Jose M Pavia pavia@uv.es
Encarnacion Alvarez encarniav@ugr.es
The Gompertz distribution with scale parameter \(\beta\), shape parameter \(\alpha\) and denoted as \(Gompertz(\beta, \alpha)\), where \(\beta>0\) and \(\alpha>0\), has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995; Rodriguez, 1977; Yee, 2022)
$$f(y)= \alpha e^{\beta y} \exp\left[ - \displaystyle \frac{\alpha}{\beta}\left(e^{\beta y} - 1 \right) \right],$$ and a cumulative distribution function given by
$$F(y)= 1 -\exp\left[ - \displaystyle \frac{\alpha}{\beta}\left(e^{\beta y} - 1 \right) \right],$$ where \(y \geq 0\).
The Gini index can be computed as
$$G = 2\left(0.5 - \displaystyle \frac{1}{E[y]}\int_{0}^{1}\int_{0}^{Q(y)}yf(y)dy\right),$$
where \(Q(y)\) is the quantile function of the Gompertz distribution, and \(E[y]\) is the expectation of the distribution. If scale is not specified it assumes the default value of 1.
Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ, USA: Wiley-Interscience.
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 1, chapter 14. Wiley, New York.
Yee, T. W. (2022). VGAM: Vector Generalized Linear and Additive Models. R package version 1.1-7, https://CRAN.R-project.org/package=VGAM.
ggamma, gbeta, gchisq, gpareto
# Gini index for the Gompertz distribution with 'scale = 1' and 'shape = 3'.
ggompertz(scale = 1, shape = 3)
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