gdagum {giniVarCI}

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

Description

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

Usage

gdagum(shape1.a, shape2.p)

Arguments

shape1.a

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

shape2.p

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

Details

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

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

where y > 0.

The Gini index can be computed as

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

where the gamma function is defined as

\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). 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 [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

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)