gfisk: Gini index for the Fisk (Log Logistic) distribution with...

In giniVarCI: Gini Indices, Variances and Confidence Intervals for Finite and Infinite Populations

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 [e-mail]

Jose M Pavia [e-mail]

Encarnacion Alvarez [e-mail]

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.

See Also

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.