Calculate the Gini index for the Pareto (I) distribution with scale parameter \(b\) and shape parameter \(s\).
gparetoI(
scale = 1,
shape = 1
)A numeric value with the Gini index. A NA is returned when a parameter is non-numeric or non-positive.
A positive real number specifying the scale parameter \(b\) of the Pareto (I) distribution. The default value is scale = 1.
A positive real number specifying the shape parameter \(s\) of the Pareto (I) 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 (I) distribution with scale parameter \(b\), shape parameter s and denoted as ParetoI(b,s), where \(b>0\) and \(s>0\), has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995; Yee, 2022)
$$f(y)= \displaystyle \frac{s}{b} \left(\frac{y}{b}\right)^{-(s+1)},$$ and a
cumulative distribution function given by
$$F(y)=1 - \displaystyle \left(\frac{y}{b}\right)^{-s},$$
where \(y>b\).
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 (I) distribution, and \(E[y]\) is the expectation of the distribution. If scale or shape are not specified they assume the default value of 1. The Pareto (I) distribution is related to the Pareto (IV) distribution: \(ParetoI(b,s) = ParetoIV(b,b,1,s)\)
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, gparetoII, gparetoIII, gparetoIV, gdagum, gburr, gfisk
# Gini index for the Pareto (I) distribution with scale 'b = 1' and shape 's = 3'.
gparetoI(scale = 1, shape = 3)
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