Entropy: Generalized entropy index

In wINEQ: Inequality Measures for Weighted Data

Generalized entropy index

Description

Computes generalized entropy index of a given variable taking into account weights.

Usage

Entropy(X, W = rep(1, length(X)), power = 0.5, zeroes = "include")

Arguments

X	is a data vector
W	is a vector of weights
power	is a entropy parameter
zeroes	defines what to do with zeroes in the data vector. Possible options are "remove" and "include". See Details for more.

Details

Entropy coefficient with respect to parameter \alpha is equal to Theil_L(X,W) whenever \alpha=0, is equal to Theil_T(X,W) whenever \alpha=1, and whenever \alpha \in (0,1) we have

GE(\alpha) = \frac{1}{\alpha(\alpha-1)W}\sum_{i=1}^{n}w_{i}\left(\left(\frac{x_{i}}{\mu}\right)^\alpha-1\right)

where W is a sum of weights and \mu is the arithmetic mean of x_{1},...,x_{n}. Entropy coefficient is not well-defined for data vector with zero values whenever parameter is zero or one. In such case, entropy index coincides with the definition of Theil L index and Theil T index, respectively, and entropy index is calculated with corresponding Theil function. Theil L always removes zeroes. Theil T enables two ways to deal with zeroes by parameter zeroes. Option "remove" discard these X's and corresponding weights. Works for power>0. Option "include" puts 0\log{0=}0 due to limiting property of p\log{p} in zero preserving zero value in dataset. It is valid only for Theil T index, that is power=0.

Value

The value of generalized entropy index

References

Shorrocks A. F.: (1980) The Class of Additively Decomposable Inequality Measures. Econometrica

Pielou E.C.: (1966) The measurement of diversity in different types of biological collections. Journal of Theoretical Biology

Examples

# Compare weighted and unweighted result
X=1:10
W=1:10
Entropy(X)
Entropy(X,W)

data(Tourism)
# Generalized entropy index for Total expenditure with sample weights
X=Tourism$Total_expenditure
W=Tourism$Sample_weight
Entropy(X,W)

wINEQ documentation built on April 23, 2026, 5:07 p.m.