gfisk {giniVarCI}R Documentation

Gini index for the Fisk (Log Logistic) distribution with user-defined shape parameters

Description

Calculates the Gini indices for the Fisk (Log Logistic) distribution with shape parameters aa (shape1.a).

Usage

gfisk(shape1.a)

Arguments

shape1.a

A vector of positive real numbers specifying shape parameters aa of the Fisk (Log Logistic) distribution.

Details

The Fisk (Log Logistic) distribution with scale parameter bb, shape parameter aa (argument shape1.a) and denoted as Fisk(b,a)Fisk(b,a), where b>0b>0 and a>0a>0, has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995; Yee, 2022)

f(y)=ay(yb)a[(yb)a+1]2,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[1+(yb)a]1,F(y)=1-\left[1 + \displaystyle \left( \frac{y}{b}\right)^{a} \right]^{-1},

where y0y \geq 0.

The Gini index can be computed as

G={1,0<a<1;1a,a1.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)Fisk(b,a) = Dagum(b,a,1).

Value

A numeric vector with the Gini indices. A NA is returned when a shape parameter is non-numeric or non-positive.

Note

The Gini index of the Fisk (Log Logistic) distribution does not depend on its scale parameter.

Author(s)

Juan F Munoz jfmunoz@ugr.es

Jose M Pavia pavia@uv.es

Encarnacion Alvarez encarniav@ugr.es

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.

See Also

gdagum, gburr, gpareto, ggompertz

Examples

# 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)

[Package giniVarCI version 0.0.1-3 Index]