wINEQ (version 1.2.1)

Theil_T: Theil T

Description

Computes Theil_T inequality measure of a given variable taking into account weights.

Usage

Theil_T(X, W = rep(1, length(X)), zeroes = "include")

Value

The value of Theil_T coefficient.

Arguments

X

is a data vector

W

is a vector of weights

zeroes

defines what to do with zeroes in the data vector. Possible options are "remove" and "include". See Details for more.

Details

Theil T index is defined as: $$T_{T} = T_{\alpha=1} = \frac{1}{N} \sum_{i=1}^N \frac{ x_{i} }{\mu} ln \big( \frac{ x_{i} }{\mu} \big)$$ where $$\mu = \frac{1}{N} \sum_{i=1}^N x_{i}$$ Formally, Theil index is defined for positive values due to logarithms. Nevertheless, in data analysis zero values may occur. There are two way we can deal with them. Option "remove" discard these X's and corresponding weights. Option "include" puts \(0\log{0=}0\) due to limiting property of \(p\log{p}\) in zero preserving zero value in dataset.

References

Serebrenik A., van den Brand M.: Theil index for aggregation of software metrics values. 26th IEEE International Conference on Software Maintenance. IEEE Computer Society.

Conceição P., Ferreira P.: (2000) The Young Person’s Guide to the Theil Index: Suggesting Intuitive Interpretations and Exploring Analytical Applications

OECD: (2020) Regions and Cities at a Glance 2020, Chapter: Indexes and estimation techniques

Examples

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

data(Tourism)
# Theil T coefficient for Total expenditure with sample weights
X=Tourism$Total_expenditure
W=Tourism$Sample_weight
Theil_T(X,W)