# gfrechet: Gini index for the Frechet distribution with user-defined... In giniVarCI: Gini Indices, Variances and Confidence Intervals for Finite and Infinite Populations

 gfrechet R Documentation

## 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 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.

### 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.