Herfindahl Index

Computes the Herfindahl (or Herfindahl-Hirschman) index as a measure of concentration or inequality.

Usage

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

Arguments

x

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

n

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

parameter

parameter \(m\) controlling sensitivity to concentration; must be strictly positive, default is 1. m = 0 is rejected: it degenerates to a constant 1 for every input.

na.rm

logical; whether to remove missing values

Value

numeric scalar containing the Herfindahl index

Details

The index is defined as the power mean of order \(m+1\) of the relative shares. For parameter = 1, the classical Herfindahl-Hirschman Index (HHI) is obtained.

Larger values indicate higher concentration. If negative values or missing values (when na.rm = FALSE) are present, NA is returned.

Note

Based on code by Achim Zeileis, rewritten to conform to package standards.

References

Cowell, F. A. (2000) Measurement of Inequality, in Atkinson, A. B., Bourguignon, F. Handbook of Income Distribution. (Eds) Amsterdam

Cowell, F. A. (1995) Measuring Inequality. Prentice Hall/Harvester Wheatshef

Hall, M., Tidemann, N. (1967) Measures of Concentration, JASA 62, 162-168.

Hirschman, A. O. (1964). The paternity of an index.

See also

gini(), atkinson()

Other diversity.concentration: entropy(), simpson()

Examples

# generate vector (of sales)
x <- c(541, 1463, 2445, 3438, 4437, 5401, 6392, 8304, 11904, 22261)

# compute Herfindahl coefficient with parameter 1
herfindahl(x)
#> [1] 0.1840812

# Some more examples
herfindahl(c(261,29,33,15,39,28,95,5,6,28,69,8,105,38,15))
#> [1] 0.1668737
herfindahl(c(783,121,112,70,201,153,425,19,37,126,325,51,442,193,41))
#> [1] 0.1301292