giniVarCI (version 0.0.1-3)

glnorm: Gini index for the Log Normal distribution with user-defined standard deviations

Description

Calculates the Gini indices for the Log Normal distribution with standard deviations \(\sigma\) (sdlog).

Usage

glnorm(sdlog)

Value

A numeric vector with the Gini indices. A NA is returned when a standard deviation is non-numeric or non-positive.

Arguments

sdlog

A vector of positive real numbers specifying standard deviations \(\sigma\) of the Log Normal distribution.

Author

Juan F Munoz

Jose M Pavia

Encarnacion Alvarez

Details

The Log Normal distribution with mean \(\mu\), standard deviation \(\sigma\) on the log scale (argument sdlog) and denoted as \(logNormal(\mu, \sigma)\), has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995) $$f(y)=\displaystyle \frac{1}{\sqrt{2\pi}\sigma y}\exp\left[- \frac{(\ln(x) - \mu)^2}{2\sigma^2} \right],$$ and a cumulative distribution function given by $$F(y)=\displaystyle \Phi\left(\frac{\ln(x) - \mu}{\sigma}\right),$$ where \(y > 0\) and $$\Phi(y) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{y} e^{-t^{2}/2}dt$$ is the cumulative distribution function of a standard Normal distribution.

The Gini index can be computed as $$G = 2\Phi\left( \displaystyle \frac{\sigma}{\sqrt{2}}\right) - 1.$$

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.

See Also

ggamma, gpareto, gchisq, gweibull

Examples

# Gini index for the Log Normal distribution with standard deviation 'sdlog = 2'.
glnorm(sdlog = 2)

# Gini indices for the Log Normal distribution with different standard deviations.
glnorm(sdlog = c(0.2, 0.5, 1:3))