giniVarCI (version 0.0.1-3)

gfrechet: Gini index for the Frechet distribution with user-defined shape parameters

Description

Calculates the Gini indices for the Frechet distribution with shape parameters $s$.

Usage

gfrechet(shape)

Value

A numeric vector with the Gini indices. A NA is returned when a shape parameter is non-numeric or smaller than 1.

Arguments

shape

A vector of positive real numbers higher or equal than 1 specifying shape parameters $s$ of the Frechet distribution.

Author

Juan F Munoz

Jose M Pavia

Encarnacion Alvarez

Details

The Frechet distribution with location parameter $a$, scale parameter $b$, shape parameter $s$ and denoted as $Frechet(a,b,s)$, where $a>0$, $b>0$ and $s>0$, has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995) $$f(y)={\displaystyle \frac{sb}{(y-a{)}^2}{\left(\frac{b}{y-a}\right)}^{s-1}\exp [-{\displaystyle {\left(\frac{b}{y-a}\right)}^s}],}$$ and a cumulative distribution function given by $$F(y)={\displaystyle \exp [-{\displaystyle {\left(\frac{b}{y-a}\right)}^s}],}$$ where $y>a$.

The Gini index, for $s\ge 1$, can be computed as $$G=2^{1/s}-1.$$

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.

See Also

gdagum, gburr, gfisk, gpareto, ggompertz

Examples

# Gini index for the Frechet distribution with a shape parameter 's = 1'.
gfrechet(shape = 1)

# Gini indices for the Frechet distribution and different shape parameters.
gfrechet(shape = 1:10)