# gfisk: Gini index for the Fisk (Log Logistic) distribution with... In giniVarCI: Gini Indices, Variances and Confidence Intervals for Finite and Infinite Populations

 gfisk R Documentation

## Gini index for the Fisk (Log Logistic) distribution with user-defined shape parameters

### Description

Calculates the Gini indices for the Fisk (Log Logistic) distribution with shape parameters a (shape1.a).

### Usage

gfisk(shape1.a)


### Arguments

 shape1.a A vector of positive real numbers specifying shape parameters a of the Fisk (Log Logistic) distribution.

### Details

The Fisk (Log Logistic) distribution with scale parameter b, shape parameter a (argument shape1.a) and denoted as Fisk(b,a), where b>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}{y}\frac{\left(\frac{y}{b}\right)^{a}}{ \left[\left(\frac{y}{b} \right)^{a} + 1 \right]^{2} },

and a cumulative distribution function given by

F(y)=1-\left[1 + \displaystyle \left( \frac{y}{b}\right)^{a} \right]^{-1},

where y \geq 0.

The Gini index can be computed as

G = \left\{ \begin{array}{cl} 1 , & 0< a <1; \\ \displaystyle \frac{1}{a}, & a \geq 1. \end{array} \right. 

The Fisk (Log Logistic) distribution is related to the Dagum distribution: Fisk(b,a) = Dagum(b,a,1).

### 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 Fisk (Log Logistic) 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.

Yee, T. W. (2022). VGAM: Vector Generalized Linear and Additive Models. R package version 1.1-7, https://CRAN.R-project.org/package=VGAM.

gdagum, gburr, gpareto, ggompertz

### Examples

# Gini index for the Fisk distribution with a shape parameter 'a = 2'.
gfisk(shape1.a = 2)

# Gini indices for the Fisk distribution and different shape parameters.
gfisk(shape1.a = 1:10)


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