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

 gweibull 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 a.

### Usage

gweibull(shape)


### Arguments

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

### Details

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

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

where y \geq 0.

The Gini index can be computed as

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.

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)


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