Rosenbluth Index

Computes the Rosenbluth index as a measure of concentration.

Usage

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

Arguments

x

numeric vector of non-negative values, such as market shares or frequencies

n

optional frequency weights. Each element of x is replicated n times.

na.rm

logical. If TRUE, missing values are removed.

Value

a numeric scalar containing the Rosenbluth index

Details

The Rosenbluth index is based on the ranked shares and is directly related to market concentration. Larger values indicate stronger concentration.

With the shares \(p_i = x_i / \sum x\) sorted in decreasing order and \(i\) their rank, the index is $$HT = 1 / (2 \sum i p_i - 1).$$ It ranges from \(1/k\) for \(k\) units of equal size to 1 for a single unit holding everything, so it is read on the same scale as the Herfindahl index rather than as an inequality measure.

If negative values or missing values (when na.rm = FALSE) are present, NA is returned. The index is undefined when all values are zero, and NA is returned in that case as well.

References

Rosenbluth, G. (1955). Measures of concentration. In: Business Concentration and Price Policy. Princeton University Press, 57-99.

Hall, M., Tideman, N. (1967). Measures of concentration. Journal of the American Statistical Association, 62, 162-168.

See also

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

Examples

# four units of equal size: the index takes its minimum 1/4
rosenbluth(c(1, 1, 1, 1))
#> [1] 0.25

# one unit holding everything: the maximum 1
rosenbluth(c(1, 0, 0, 0))
#> [1] 1

# a dominant unit next to three small ones
rosenbluth(c(10, 1, 1, 1))
#> [1] 0.52

# frequency weights replicate the values
rosenbluth(c(10, 1), n = c(1, 3))
#> [1] 0.52