Calculates the Gini index for the Pareto (IV) distribution with location parameter \(a\), scale parameter \(b\), inequality parameter \(g\) and shape parameter \(s\).
gparetoIV(
location = 0,
scale = 1,
inequality = 1,
shape = 1
)A numeric value with the Gini index. A NA is returned when a parameter is non-numeric or positive, except for the location parameter that can be equal to 0.
A non-negative real number specifying the location parameter \(a\) of the Pareto (IV) distribution. The default value is location = 0.
A positive real number specifying the scale parameter \(b\) of the Pareto (IV) distribution. The default value is scale = 1.
A positive real number specifying the inequality parameter \(g\) of the Pareto (IV) distribution. The default value is inequality = 1.
A positive real number specifying the shape parameter \(s\) of the Pareto (IV) distribution. The default value is shape = 1.
Juan F Munoz jfmunoz@ugr.es
Jose M Pavia pavia@uv.es
Encarnacion Alvarez encarniav@ugr.es
The Pareto (IV) distribution with location parameter \(a\), scale parameter \(b\), inequality parameter \(g\), shape parameter \(s\) and denoted as ParetoIV(a,b,g,s), where \(a \geq 0\), \(b>0\), \(g>0\) and \(s>0\), has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995; Yee, 2022)
$$f(y)= \displaystyle \frac{s}{bg} \left( \frac{y-a}{b}\right)^{1/g-1} \left[1 + \left( \frac{y-a}{b}\right)^{1/g} \right]^{-(s+1)},$$ and a cumulative distribution function given by
$$F(y)=1- \left[1 + \displaystyle \left( \frac{y-a}{b}\right)^{1/g} \right]^{-s},$$
where \(y>a\).
The Gini index can be computed as
$$G = 2\left(0.5 - \displaystyle \frac{1}{E[y]}\int_{0}^{1}\int_{0}^{Q(y)}yf(y)dy\right),$$
where \(Q(y)\) is the quantile function of the Pareto (IV) distribution, and \(E[y]\) is the expectation of the distribution. If location is not specified it assumes the default value of 0, and the remaining parameters assume the default value of 1. The Pareto (IV) distribution is related to:
1. The Burr distribution: \(ParetoIV(0,b,g,s) = BurrXII(b,1/g,s)\).
2. The Pareto (I) distribution: \(ParetoIV(b,b,1,s) = ParetoI(b,s)\).
3. The Pareto (II) distribution: \(ParetoIV(a,b,1,s) = ParetoII(a,b,s)\).
4. The Pareto (III) distribution: \(ParetoIV(a,b,g,1) = ParetoIII(a,b,g)\).
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.
gpareto, gparetoI, gparetoII, gparetoIII, gdagum, gburr, gfisk
# Gini index for the Pareto (IV) distribution with 'a = 1', 'b = 1', 'g = 0.5', 's = 1'.
gparetoIV(location = 1, scale = 1, inequality = 0.5, shape = 1)
# Gini index for the Pareto (IV) distribution with 'a = 1', 'b = 1', 'g = 2', 's = 3'.
gparetoIV(location = 1, scale = 1, inequality = 2, shape = 3)
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