Calculates the Gini index for the Dagum distribution with shape parameters \(a\) (shape1.a) and \(p\) (shape2.p).
gdagum(shape1.a, shape2.p)A numeric value with the Gini index. A NA is returned when a shape parameter is non-numeric or non-positive.
A positive real number specifying the shape1 parameter \(a\) of the Dagum distribution.
A positive real number specifying the shape parameter \(p\) of the Dagum distribution.
Juan F Munoz jfmunoz@ugr.es
Jose M Pavia pavia@uv.es
Encarnacion Alvarez encarniav@ugr.es
The Dagum distribution with scale parameter \(b\), shape parameters \(a\) (argument shape1.a) and \(p\) (argument shape2.p) and denoted as \(Dagum(b,a,p)\) , where \(b>0\), \(a>0\) and \(p>0\),
has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995; Rodriguez, 1977; Yee, 2022)
$$f(y) = \displaystyle \frac{ap}{y}\frac{\left(\frac{y}{b}\right)^{ap}}{ \left[\left(\frac{y}{b} \right)^{a} + 1 \right]^{p+1} },$$
and a cumulative distribution function given by
$$F(y)= \left[1 + \displaystyle \left( \frac{y}{b}\right)^{-a} \right]^{-p},$$
where \(y > 0\).
The Gini index can be computed as $$G = \displaystyle \frac{\Gamma(p)\Gamma(2p+1/a)}{\Gamma(2p)\Gamma(p+1/a)}-1,$$ where the gamma function is defined as $$\Gamma(\alpha) = \int_{0}^{\infty}t^{\alpha-1}e^{-t}dt.$$ The Dagum distribution is also known the Burr III, inverse Burr, beta-K, or 3-parameter kappa distribution. The Dagum distribution is related to the Fisk (Log Logistic) distribution: \(Dagum(b,a,1) = Fisk(b,a)\). The Dagum distribution is also related to the inverse Lomax distribution and the inverse paralogistic distribution (see Kleiber and Kotz, 2003; Johnson et al., 1995; Yee, 2022).
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.
gburr, gpareto, gfisk, ggompertz, gfrechet
# Gini index for the Dagum distribution with shape parameters 'a = 2' and 'p = 20'.
gdagum(shape1.a = 2, shape2.p = 20)
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