Gini: Gini Coefficient

Description

Compute the Gini coefficient.

Usage

Gini(x, n = rep(1, length(x)), unbiased = TRUE, 
     conf.level = NA, R = 1000, type = "bca", na.rm = FALSE)

Arguments

a vector containing at least non-negative elements.

a vector of frequencies (weights), must be same length as x.

unbiased

logical. In order for G to be an unbiased estimate of the true population value, calculated gini is multiplied by n/(n-1). Default is TRUE. (See Dixon, 1987)

conf.level

confidence level for the returned confidence interval, restricted to lie between 0 and 1. If set to TRUE the bootstrap confidence intervals are calculated. If set to NA, which is the default, no confidence intervals are returned.

The number of bootstrap replicates. Usually this will be a single positive integer. For importance resampling, some resamples may use one set of weights and others use a different set of weights. In this case R would be a vector of integers where

type

A vector of character strings representing the type of intervals required. The value should be any subset of the values c("norm","basic", "stud", "perc", "bca") or simply "

na.rm

logical. Should missing values be removed? Defaults to FALSE.

Value

If conf.level is NA then the result will be a single numeric value. If conf.level is provided the result will be a vector with 3 elements for estimate, lower confidence intervall and upper for the upper one.

Details

The small sample variance properties of the Gini coefficient are not known, and large sample approximations to the variance of the coefficient are poor (Mills and Zandvakili, 1997; Glasser, 1962; Dixon et al., 1987), therefore confidence intervals are calculated via bootstrap re-sampling methods (Efron and Tibshirani, 1997). Two types of bootstrap confidence intervals are commonly used, these are percentile and bias-corrected (Mills and Zandvakili, 1997; Dixon et al., 1987; Efron and Tibshirani, 1997). The bias-corrected intervals are most appropriate for most applications. This is set as default for the type argument ("bca"). Dixon (1987) describes a refinement of the bias-corrected method known as 'accelerated' - this produces values very closed to conventional bias corrected intervals. (Iain Buchan (2002) Calculating the Gini coefficient of inequality, see: http://www.statsdirect.com/help/nonparametric_methods/gini_coefficient.htm)

References

Cowell, F. A. (2000) Measurement of Inequality in Atkinson, A. B. / Bourguignon, F. (Eds): Handbook of Income Distribution. Amsterdam. Cowell, F. A. (1995) Measuring Inequality Harvester Wheatshef: Prentice Hall. Marshall, Olkin (1979) Inequalities: Theory of Majorization and Its Applications. New York: Academic Press. Glasser C. (1962) Variance formulas for the mean difference and coefficient of concentration. Journal of the American Statistical Association 57:648-654. Mills JA, Zandvakili A. (1997). Statistical inference via bootstrapping for measures of inequality. Journal of Applied Econometrics 12:133-150. Dixon, PM, Weiner J., Mitchell-Olds T, Woodley R. (1987) Boot-strapping the Gini coefficient of inequality. Ecology 68:1548-1551. Efron B, Tibshirani R. (1997) Improvements on cross-validation: The bootstrap method. Journal of the American Statistical Association 92:548-560.

Examples

Run this code

# generate vector (of incomes)
x <- c(541, 1463, 2445, 3438, 4437, 5401, 6392, 8304, 11904, 22261)

# compute Gini coefficient
Gini(x)

# working with weights
fl <- c(2.5,7.5,15,35,75,150)    # midpoints of classes
n  <- c(25,13,10,5,5,2)          # frequencies

Gini(fl, n, conf.level=0.95, unbiased=FALSE)

# some special cases
x <- c(10,10,0,0,0)
plot(Lc(x))

Gini(x, unbiased=FALSE)

# the same with weights
Gini(x=c(10,0), n=c(2,3), unbiased=FALSE)

# perfect balance
Gini(c(10,10,10))

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