gparetoIII {giniVarCI}

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
)

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.

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).

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].

Note

The Gini index of the Pareto (III) distribution does not depend on its location and scale parameters.

Author(s)

Juan F Munoz [e-mail]

Jose M Pavia [e-mail]

Encarnacion Alvarez [e-mail]

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))