The Fisk (Log Logistic) distribution with scale parameter \(b\), shape parameter \(a\) (argument shape1.a) and denoted as \(Fisk(b,a)\), where \(b>0\) and \(a>0\), has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995; Yee, 2022)
$$f(y) = \displaystyle \frac{a}{y}\frac{\left(\frac{y}{b}\right)^{a}}{ \left[\left(\frac{y}{b} \right)^{a} + 1 \right]^{2} },$$ and a cumulative distribution function given by
$$F(y)=1-\left[1 + \displaystyle \left( \frac{y}{b}\right)^{a} \right]^{-1},$$
where \(y \geq 0\).
The Gini index can be computed as
$$G = \left\{
\begin{array}{cl}
1 , & 0< a <1; \\
\displaystyle \frac{1}{a}, & a \geq 1.
\end{array}
\right.
$$ The Fisk (Log Logistic) distribution is related to the Dagum distribution: \(Fisk(b,a) = Dagum(b,a,1)\).
References
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.
# Gini index for the Fisk distribution with a shape parameter 'a = 2'.gfisk(shape1.a = 2)
# Gini indices for the Fisk distribution and different shape parameters.gfisk(shape1.a = 1:10)