glnorm {giniVarCI}

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)

Arguments

sdlog

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

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.

Value

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

Note

The Gini index of the logNormal distribution does not depend on the mean 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.

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))