gdagum {giniVarCI}R Documentation

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

Description

Calculates the Gini index for the Dagum distribution with shape parameters aa (shape1.a) and pp (shape2.p).

Usage

gdagum(shape1.a, shape2.p)

Arguments

shape1.a

A positive real number specifying the shape1 parameter aa of the Dagum distribution.

shape2.p

A positive real number specifying the shape parameter pp of the Dagum distribution.

Details

The Dagum distribution with scale parameter bb, shape parameters aa (argument shape1.a) and pp (argument shape2.p) and denoted as Dagum(b,a,p)Dagum(b,a,p) , where b>0b>0, a>0a>0 and p>0p>0, has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995; Rodriguez, 1977; Yee, 2022)

f(y)=apy(yb)ap[(yb)a+1]p+1,f(y) = \displaystyle \frac{ap}{y}\frac{\left(\frac{y}{b}\right)^{ap}}{ \left[\left(\frac{y}{b} \right)^{a} + 1 \right]^{p+1} },

and a cumulative distribution function given by

F(y)=[1+(yb)a]p,F(y)= \left[1 + \displaystyle \left( \frac{y}{b}\right)^{-a} \right]^{-p},

where y>0y > 0.

The Gini index can be computed as

G=Γ(p)Γ(2p+1/a)Γ(2p)Γ(p+1/a)1,G = \displaystyle \frac{\Gamma(p)\Gamma(2p+1/a)}{\Gamma(2p)\Gamma(p+1/a)}-1,

where the gamma function is defined as

Γ(α)=0tα1etdt.\Gamma(\alpha) = \int_{0}^{\infty}t^{\alpha-1}e^{-t}dt.

The Dagum distribution is also known the Burr III, inverse Burr, beta-K, or 3-parameter kappa distribution. The Dagum distribution is related to the Fisk (Log Logistic) distribution: Dagum(b,a,1)=Fisk(b,a)Dagum(b,a,1) = Fisk(b,a). The Dagum distribution is also related to the inverse Lomax distribution and the inverse paralogistic distribution (see Kleiber and Kotz, 2003; Johnson et al., 1995; Yee, 2022).

Value

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

Note

The Gini index of the Dagum distribution does not depend on its scale parameter.

Author(s)

Juan F Munoz jfmunoz@ugr.es

Jose M Pavia pavia@uv.es

Encarnacion Alvarez encarniav@ugr.es

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

gburr, gpareto, gfisk, ggompertz, gfrechet

Examples

# Gini index for the Dagum distribution with shape parameters 'a = 2' and 'p = 20'.
gdagum(shape1.a = 2, shape2.p = 20)


[Package giniVarCI version 0.0.1-3 Index]