ggompertz {giniVarCI}

Gini index for the Gompertz distribution with user-defined scale and shape parameters

Description

Calculate the Gini index for the Gompertz distribution with scale parameter \beta and shape parameter \alpha.

Usage

ggompertz(
 scale = 1,
 shape
)

Arguments

scale

A positive real number specifying the scale parameter \beta of the Gompertz distribution. The default value is scale = 1.

shape

A positive real number specifying the shape parameter \alpha of the Gompertz distribution.

Details

The Gompertz distribution with scale parameter \beta, shape parameter \alpha and denoted as Gompertz(\beta, \alpha), where \beta>0 and \alpha>0, has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995; Rodriguez, 1977; Yee, 2022)

f(y)= \alpha e^{\beta y} \exp\left[ - \displaystyle \frac{\alpha}{\beta}\left(e^{\beta y} - 1 \right) \right],

and a cumulative distribution function given by

F(y)= 1 -\exp\left[ - \displaystyle \frac{\alpha}{\beta}\left(e^{\beta y} - 1 \right) \right],

where y \geq 0.

The Gini index can be computed as

G = 2\left(0.5 - \displaystyle \frac{1}{E[y]}\int_{0}^{1}\int_{0}^{Q(y)}yf(y)dy\right),

where Q(y) is the quantile function of the Gompertz distribution, and E[y] is the expectation of the distribution. If scale is not specified it assumes the default value of 1.

Value

A numeric value with the Gini index. A NA is returned when a parameter is non-numeric or non-positive.

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

ggamma, gbeta, gchisq, gpareto

Examples

# Gini index for the Gompertz distribution with 'scale = 1' and 'shape = 3'.
ggompertz(scale = 1, shape = 3)