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 numeric 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 \((x_1,\dots,x_n)\).

Time complexity: \(O(n)\) for sorted (increasingly) data. Otherwise, the vector will be sorted.

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.

Gagolewski M., Bartoszuk M., Cena A., Genie: A new, fast, and outlier-resistant hierarchical clustering algorithm, Information Sciences 363, 2016, pp. 8-23. doi:10.1016/j.ins.2016.05.003

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. 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