Atkinson Index

Computes the Atkinson inequality index.

Usage

atkinson(
  x,
  n = rep(1, length(x)),
  epsilon = 0.5,
  na.rm = FALSE,
  tol = 0.00000001
)

Arguments

x: numeric vector of non-negative values, such as incomes
n: optional frequency weights; either a single non-negative whole number or a vector having the same length as x
epsilon: single non-negative numeric value specifying the inequality aversion parameter
na.rm: logical; whether missing values in x are removed
tol: single non-negative numeric value specifying the tolerance for treating epsilon as equal to one

Value

a numeric value in the interval [0, 1], or NA_real_ if the index is undefined

Details

With frequency weights $n_i$, the weighted arithmetic mean is

$$ \bar{x}_n = \frac{\sum_i n_i x_i}{\sum_i n_i}. $$

For $\varepsilon \ne 1$, the Atkinson index is

$$ A(\varepsilon) = 1 - \frac{ \left( \frac{\sum_i n_i x_i^{1-\varepsilon}} {\sum_i n_i} \right)^{1/(1-\varepsilon)} }{\bar{x}_n}. $$

For $\varepsilon = 1$,

$$ A(1) = 1 - \frac{ \exp\left( \frac{\sum_i n_i \log(x_i)} {\sum_i n_i} \right) }{\bar{x}_n}. $$

The calculation uses normalized frequency weights and logarithmic power means. It therefore does not construct the potentially very large vector that would result from rep(x, n).

Observations with zero frequency are ignored. If all frequencies are zero or no observations remain after removing missing values, NA_real_ is returned.

If all values are zero, the index is defined as zero. If at least one value is zero and epsilon >= 1, the equally distributed equivalent value is zero and the index is one.

Negative values, non-finite values, and missing values when na.rm = FALSE produce NA_real_. A negative epsilon also produces NA_real_.

References

Atkinson, A. B. (1970). On the measurement of inequality. Journal of Economic Theory, 2(3), 244–263.

Examples

x <- c(541, 1463, 2445, 3438, 4437,
       5401, 6392, 8304, 11904, 22261)

atkinson(x)
#> [1] 0.1796591
atkinson(x, epsilon = 1)
#> [1] 0.3518251
atkinson(x, epsilon = 2)
#> [1] 0.6290111

# frequency weights
atkinson(c(10, 20, 30), n = c(3, 1, 1))
#> [1] 0.05558586

# zero incomes
atkinson(c(0, 10, 20), epsilon = 1)
#> [1] 1