R: Gini Coefficient

Gini {DescTools} | R Documentation

## Gini Coefficient

### Description

Compute the Gini coefficient, the most commonly used measure of inequality.

### Usage

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

### Arguments

`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.

### Details

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: http://www.statsdirect.com/help/default.htm#nonparametric_methods/gini.htm)

### Value

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

### Author(s)

Andri Signorell <andri@signorell.net>

### 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.

### See Also

See `Herfindahl`, `Rosenbluth` for concentration measures, `Lc` for the Lorenz curve
`ineq()` in the package ineq contains additional inequality measures

### Examples

    # 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))

[Package DescTools version 0.99.28 Index]