gparetoI {giniVarCI}

Gini index for the Pareto (I) distribution with user-defined scale and shape parameters

Description

Calculate the Gini index for the Pareto (I) distribution with scale parameter b and shape parameter s.

Usage

gparetoI(
 scale = 1,
 shape = 1
)

Arguments

scale

A positive real number specifying the scale parameter b of the Pareto (I) distribution. The default value is scale = 1.

shape

A positive real number specifying the shape parameter s of the Pareto (I) distribution. The default value is shape = 1.

Details

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)

Value

A numeric value with the Gini index. A NA is returned when a parameter is non-numeric or non-positive.

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, gparetoII, gparetoIII, gparetoIV, gdagum, gburr, gfisk

Examples

# Gini index for the Pareto (I) distribution with scale 'b = 1' and shape 's = 3'.
gparetoI(scale = 1, shape = 3)