gweibull {giniVarCI}R Documentation

Gini index for the Weibull distribution with user-defined shape parameters

Description

Calculate the Gini indices for the Weibull distribution with shape parameters aa.

Usage

gweibull(shape)

Arguments

shape

A vector of positive real numbers specifying shape parameters aa of the Weibull distribution.

Details

The Weibull distribution with scale parameter σ\sigma, shape parameter aa, and denoted as Weibull(σ,a)Weibull(\sigma, a), where σ>0\sigma>0 and a>0a>0, has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995; Yee, 2022)

f(y)=aσ(yσ)a1e(y/σ)a,f(y) = \displaystyle \frac{a}{\sigma}\left(\frac{y}{\sigma}\right)^{a-1}e^{-(y/\sigma)^{a}},

and a cumulative distribution function given by

F(y)=1e(y/σ)a,F(y) = \displaystyle 1 - e^{-(y/\sigma)^{a}},

where y0y \geq 0.

The Gini index can be computed as

G=121/a.G = 1-2^{-1/a}.

Value

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

Note

The Gini index of the Weibull distribution does not depend on its scale parameter.

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

gbeta, ggamma, gchisq, gunif

Examples

# Gini index for the Weibull distribution with 'shape = 1'.
gweibull(shape = 1)

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

[Package giniVarCI version 0.0.1-3 Index]