inequality: Inequality Measures¶

Description¶

gini_index() gives the normalised Gini index, bonferroni_index() implements the Bonferroni index, and devergottini_index() implements the De Vergottini index.

Usage¶

gini_index(x)

bonferroni_index(x)

devergottini_index(x)

Arguments¶


`x`	numeric vector of non-negative values

Details¶

These indices can be used to quantify the “inequality” of a sample. They can be conceived as normalised measures of data dispersion. For constant vectors (perfect equity), the indices yield values of 0. Vectors with all elements but one equal to 0 (perfect inequality), are assigned scores of 1. They follow the Pigou-Dalton principle (are Schur-convex): setting \(x_i = x_i - h\) and \(x_j = x_j + h\) with \(h > 0\) and \(x_i - h \geq x_j + h\) (taking from the “rich” and giving to the “poor”) decreases the inequality.

These indices have applications in economics, amongst others. The Genie clustering algorithm uses the Gini index as a measure of the inequality of cluster sizes.

The normalised Gini index is given by:

\[ G(x_1,\dots,x_n) = \frac{ \sum_{i=1}^{n} (n-2i+1) x_{\sigma(n-i+1)} }{ (n-1) \sum_{i=1}^n x_i }.\]

The normalised Bonferroni index is given by:

\[ B(x_1,\dots,x_n) = \frac{ \sum_{i=1}^{n} (n-\sum_{j=1}^i \frac{n}{n-j+1}) x_{\sigma(n-i+1)} }{ (n-1) \sum_{i=1}^n x_i }.\]

The normalised De Vergottini index is given by:

\[ V(x_1,\dots,x_n) = \frac{1}{\sum_{i=2}^n \frac{1}{i}} \left( \frac{ \sum_{i=1}^n \left( \sum_{j=i}^{n} \frac{1}{j}\right) x_{\sigma(n-i+1)} }{\sum_{i=1}^{n} x_i} - 1 \right).\]

Here, \(\sigma\) is an ordering permutation of \((x_1,\dots,x_n)\).

Value¶

The value of the inequality index, a number in \([0, 1]\).

Author(s)¶

Marek Gagolewski and other contributors

References¶

Bonferroni, C., Elementi di Statistica Generale, Libreria Seber, Firenze, 1930.

Gini, C., Variabilita e Mutabilita, Tipografia di Paolo Cuppini, Bologna, 1912.

Examples¶

gini_index(c(2, 2, 2, 2, 2))   # no inequality
## [1] 0
gini_index(c(0, 0, 10, 0, 0))  # one has it all
## [1] 1
gini_index(c(7, 0, 3, 0, 0))   # give to the poor, take away from the rich
## [1] 0.85
gini_index(c(6, 0, 3, 1, 0))   # (a.k.a. the Pigou-Dalton principle)
## [1] 0.75
bonferroni_index(c(2, 2, 2, 2, 2))
## [1] 0
bonferroni_index(c(0, 0, 10, 0, 0))
## [1] 1
bonferroni_index(c(7, 0, 3, 0, 0))
## [1] 0.90625
bonferroni_index(c(6, 0, 3, 1, 0))
## [1] 0.8333333
devergottini_index(c(2, 2, 2, 2, 2))
## [1] 0
devergottini_index(c(0, 0, 10, 0, 0))
## [1] 1
devergottini_index(c(7, 0, 3, 0, 0))
## [1] 0.7662338
devergottini_index(c(6, 0, 3, 1, 0))
## [1] 0.6493506