Gini Coefficient
Compute the Gini coefficient, the most commonly used measure of inequality.
Gini(x, n = rep(1, length(x)), unbiased = TRUE,
conf.level = NA, R = 1000, type = "bca", na.rm = FALSE)x |
a vector containing at least non-negative elements. The result will be |
n |
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 confidence interval, restricted to lie between 0 and 1.
If set to |
R |
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 each component gives the number of resamples from each of the rows of weights. |
type |
character string representing the type of interval required.
The value should be one out of the c( |
na.rm |
logical. Should missing values be removed? Defaults to FALSE. |
The range of the Gini coefficient goes from 0 (no concentration) to √(\frac{n-1}{n}) (maximal concentration). The bias corrected Gini coefficient goes from 0 to 1.
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: https://www.statsdirect.com/help/default.htm#nonparametric_methods/gini.htm)
If conf.level is set to NA then the result will be
|
a single numeric value |
and
if a conf.level is provided, a named numeric vector with 3 elements:
gini |
Gini coefficient |
lwr.ci |
lower bound of the confidence interval |
upr.ci |
upper bound of the confidence interval |
Andri Signorell <andri@signorell.net>
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.
See Herfindahl, Rosenbluth for concentration measures,
Lc for the Lorenz curveineq() in the package ineq contains additional inequality measures
# 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 # with confidence intervals 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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