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

 ggamma R Documentation

## Gini index for the Gamma distribution with user-defined shape parameter

### Description

Calculates the Gini indices for the Gamma distribution with shape parameters \alpha.

### Usage

ggamma(shape)


### Arguments

 shape A vector of positive real numbers specifying the shape parameters \alpha of the Gamma distribution.

### Details

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

f(y) = \displaystyle \frac{1}{\sigma^{\alpha}\Gamma(\alpha)}y^{\alpha-1}e^{-y/\sigma},

and a cumulative distribution function given by

F(y) = \frac{\gamma\left(\alpha, \frac{y}{\sigma}\right)}{\Gamma(\alpha)},

where y \geq 0, the gamma function is defined by

\Gamma(\alpha) = \int_{0}^{\infty}t^{\alpha-1}e^{-t}dt,

and the lower incomplete gamma function is given by

\gamma(\alpha,y) = \int_{0}^{y}t^{\alpha-1}e^{-t}dt.

The Gini index can be computed as

G = \displaystyle \frac{\Gamma\left(\frac{2\alpha+1}{2}\right)}{\alpha\Gamma(\alpha)\sqrt{\pi}}.

The Gamma distribution is related to the Chi-squared distribution: Gamma(n/2, 2) = \chi_{n}^2.

### 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 Gamma 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.

gchisq, gf, gbeta, gweibull, glnorm

### Examples

# Gini index for the Gamma distribution with 'shape = 1'.
ggamma(shape = 1)

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


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