giniVarCI (version 0.0.1-3)

gburr: Gini index for the Burr Type XII (Singh-Maddala) distribution with user-defined scale and shape parameters

Description

Calculates the Gini index for the Burr Type XII (Singh-Maddala) distribution with scale parameter \(b\) and shape parameters \(g\) (shape.g) and \(s\) (shape.s).

Usage

gburr(
 scale = 1,
 shape.g = 1,
 shape.s = 1
)

Value

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

Arguments

scale

A positive real number specifying the scale parameter \(b\) of the Burr Type XII (Singh-Maddala) distribution. The default value is scale = 1.

shape.g

A positive real number specifying the shape parameter \(g\) of the Burr Type XII (Singh-Maddala) distribution. The default value is shape.g = 1.

shape.s

A positive real number specifying the shape parameter \(s\) of the Burr Type XII (Singh-Maddala) distribution. The default value is shape.s = 1.

Author

Juan F Munoz

Jose M Pavia

Encarnacion Alvarez

Details

The Burr Type XII (Singh-Maddala) distribution with scale parameter \(b\), shape parameters \(g\) (argument shape.g) and \(s\) (argument shape.s) and denoted as \(BurrXII(b,g,s)\), where \(b>0\), \(g>0\) and \(s>0\), has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995; Rodriguez, 1977; Yee, 2022) $$f(y) = \displaystyle \frac{gs}{b}\left(\frac{y}{b}\right)^{g-1}\left[1 + \left(\frac{y}{b}\right)^{g}\right]^{-(s+1)},$$ and a cumulative distribution function given by $$F(y)=1-\left[1 + \displaystyle \left( \frac{y}{b}\right)^{g} \right]^{-s},$$ where \(y>0\).

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 Burr Type XII (Singh-Maddala) distribution, and \(E[y]\) is the expectation of the distribution. The Burr Type XII (Singh-Maddala) distribution is related to the Pareto (IV) distribution: \(BurrXII(b,g,s) = ParetoIV(0,b,1/g,s)\).

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.

Rodriguez, R. N. (1977). A guide to the Burr type XII distributions. Biometrika, 64(1), 129-134.

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

gparetoIV, gpareto, gparetoI, gparetoII, gparetoIII, gfisk

Examples

# Gini index for the Burr Type XII distribution with 'scale = 1', 'shape.g = 2', 'shape.s = 1'.
gburr(scale = 1, shape.g = 2, shape.s = 1)

# Gini index for the Burr Type XII distribution with 'scale = 1', 'shape.g = 5', 'shape.s = 3'.
gburr(scale = 1, shape.g = 5, shape.s = 3)