Theil Index

Computes the Theil inequality index (Theil T).

Usage

theil(x, n = rep(1, length(x)), na.rm = FALSE)

Arguments

x

numeric vector of non-negative values, such as incomes

n

optional frequency weights. Each element of x is replicated n times. Must be a vector of non-negative integers of the same length as x.

na.rm

logical. If TRUE, missing values are removed.

Value

a numeric scalar containing the Theil index. The value is 0 under perfect equality and increases with inequality, up to a maximum of \(\log(n)\), attained when a single unit holds the entire total.

Details

The Theil index is an entropy-based measure of inequality. It belongs to the class of Generalized Entropy measures with parameter \(\alpha = 1\).

The Theil T index is defined as

$$ T = \frac{1}{n} \sum \frac{x_i}{\bar{x}} \log\left(\frac{x_i}{\bar{x}}\right) $$

where \(\bar{x}\) is the mean of x and \(n\) the number of (replicated) observations.

Zero values are admissible: following the usual convention \(x \log x \to 0\) as \(x \to 0\), they contribute 0 to the sum.

The index is decomposable into within- and between-group components, which makes it particularly useful in applied inequality analysis.

If negative values or missing values (when na.rm = FALSE) are present, NA is returned. The same holds if no observation remains after removing missing values.

References

Theil, H. (1967). Economics and Information Theory.

See also

Other inequality: atkinson(), divCoef(), gini(), lc(), rosenbluth()

Examples

theil(c(10, 10, 10, 10))          # perfect equality: 0
#> [1] 0
theil(c(0, 0, 0, 40))             # everything in one hand: log(4)
#> [1] 1.386294
theil(c(1, 2, 3, 4, 5))
#> [1] 0.1196876

# frequency weights replicate the observations
theil(1:3, n = c(1, 2, 3))
#> [1] 0.05699495
theil(rep(1:3, times = c(1, 2, 3)))
#> [1] 0.05699495