giniVarCI (version 0.0.1-3)

gparetoIII: Gini index for the Pareto (III) distribution with user-defined inequality parameters

Description

Calculate the Gini index for the Pareto (III) distribution with inequality parameters \(g\).

Usage

gparetoIII(
 inequality = 1
)

Value

A numeric vector with the Gini indices. A NA is returned when a inequality parameter is non-numeric or it is out of the interval \([0,1]\).

Arguments

inequality

A vector of positive numbers in the \([0,1]\) interval specifying inequality parameters \(g\) of the Pareto (III) distribution. The default value is inequality = 1.

Author

Juan F Munoz

Jose M Pavia

Encarnacion Alvarez

Details

The Pareto (III) distribution with location parameter \(a\), scale parameter \(b\), inequality parameter g and denoted as \(ParetoIII(a,b,g)\), where \(a>0\), \(b>0\), and \(g \in [0,1]\), has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995; Yee, 2022) $$f(y)= \displaystyle \frac{1}{bg} \left( \frac{y-a}{b}\right)^{1/g-1} \left[1 + \left( \frac{y-a}{b}\right)^{1/g} \right]^{-2},$$ and a cumulative distribution function given by $$F(y)=1-\left[1 + \displaystyle \left( \frac{y-a}{b}\right)^{1/g} \right]^{-1},$$ where \(y>a\).

The Gini index is \(G = g.\)

If inequality is not specified it assumes the default value of 1. The Pareto (III) distribution is related to the Pareto (IV) distribution: \(ParetoIII(a,b,g) = ParetoIV(a,b,g,1)\).

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.

See Also

gpareto, gparetoI, gparetoII, gparetoIV, gdagum, gburr, gfisk

Examples

# Gini index for the Pareto (III) distribution with inequality parameter 'g = 0.3'.
gparetoIII(inequality = 0.3)

# Gini indices for the Pareto (III) distribution with different inequality parameters.
gparetoIII(inequality = seq(0.1, 0.9, by=0.1))