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 `NA`

, if x contains negative elements.

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 `TRUE`

the bootstrap confidence intervals are calculated.
If set to `NA`

(default) no confidence intervals are returned.

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. This is ignored if no confidence intervals are to be calculated.

type

character string representing the type of interval required.
The value should be one out of the c(`"norm"`

,`"basic"`

, `"stud"`

,
`"perc"`

or `"bca"`

).
This argument is ignored if no confidence intervals are to be calculated.

na.rm

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

If `conf.level`

is set to `NA`

then the result will be

a single numeric value

Gini coefficient

lower bound of the confidence interval

upper bound of the confidence interval

The range of the Gini coefficient goes from 0 (no concentration) to \(\sqrt(\frac{n-1}{n})\) (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)

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 curve
`ineq()`

in the package ineq contains additional inequality measures

# NOT RUN { # 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)) # }