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

In giniVarCI: Gini Indices, Variances and Confidence Intervals for Finite and Infinite Populations

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)

Arguments

shape

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

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\left[- \displaystyle \left(\frac{b}{y-a}\right)^{s} \right],

and a cumulative distribution function given by

F(y)= \displaystyle \exp\left[- \displaystyle \left(\frac{b}{y-a}\right)^{s} \right],

where y > a.

The Gini index, for s \geq 1, can be computed as

G = 2^{1/s} -1.

Value

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

Note

The Gini index of the Frechet distribution does not depend on its location and scale parameters and only is defined when its shape parameter is at least 1.

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.

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)

giniVarCI documentation built on May 29, 2024, 3:36 a.m.