gpareto {giniVarCI}

Gini index for the Pareto distribution with user-defined shape parameters

Description

Calculates the Gini indices for the Pareto distribution with shape parameters \alpha.

Usage

gpareto(shape)

Arguments

shape

A vector of positive real numbers specifying shape parameters \alpha of the Pareto distribution.

Details

The Pareto distribution with scale parameter k, shape parameter \alpha and denoted as Pareto(k, \alpha), where k>0 and \alpha>0, has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995; Yee, 2022)

f(y)=\displaystyle \frac{\alpha k^{\alpha}}{y^{\alpha +1}},

and a cumulative distribution function given by

F(y) = \displaystyle 1 - \left(\frac{k}{y}\right)^{\alpha},

where y \geq k.

The Gini index can be computed as

G = \left\{ \begin{array}{cl} 1 , & 0<\alpha <1; \\ \displaystyle \frac{1}{2\alpha-1}, & \alpha \geq 1. \end{array} \right.

Value

A numeric vector with the Gini indices. A NA is returned when a shape parameter is non-numeric or non-positive.

Note

The Gini index of the Pareto distribution does not depend on the shape parameter.

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

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

Examples

# Gini index for the Pareto distribution with 'shape = 2'.
gpareto(shape = 2)

# Gini indices for the Pareto distribution and different shape parameters.
gpareto(shape = 1:5)