# glnorm: Gini index for the Log Normal distribution with user-defined... In giniVarCI: Gini Indices, Variances and Confidence Intervals for Finite and Infinite Populations

 glnorm R Documentation

## 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 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.

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))


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