gfrechet {giniVarCI} | 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 |
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)