gfisk | R Documentation |
Calculates the Gini indices for the Fisk (Log Logistic) distribution with shape parameters a
(shape1.a
).
gfisk(shape1.a)
shape1.a |
A vector of positive real numbers specifying shape parameters |
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)
.
A numeric vector with the Gini indices. A NA
is returned when a shape parameter is non-numeric or non-positive.
The Gini index of the Fisk (Log Logistic) distribution does not depend on its scale parameter.
Juan F Munoz jfmunoz@ugr.es
Jose M Pavia pavia@uv.es
Encarnacion Alvarez encarniav@ugr.es
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.
gdagum
, gburr
, gpareto
, ggompertz
# 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)
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