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Concept
Theil index
Summary
The Theil index is a statistic primarily used to measure economic inequality and other economic phenomena, though it has also been used to measure racial segregation.
The Theil index TT is the same as redundancy in information theory which is the maximum possible entropy of the data minus the observed entropy. It is a special case of the generalized entropy index. It can be viewed as a measure of redundancy, lack of diversity, isolation, segregation, inequality, non-randomness, and compressibility. It was proposed by a Dutch econometrician Henri Theil (1924-2000) at the Erasmus University Rotterdam.
Henri Theil himself said (1967): "The (Theil) index can be interpreted as the expected information content of the indirect message which transforms the population shares as prior probabilities into the income shares as posterior probabilities."
Amartya Sen noted, "But the fact remains that the Theil index is an arbitrary formula, and the average of the logarithms of the reciprocals of income shares weighted by income is not a measure that is exactly overflowing with intuitive sense."
For a population of N "agents" each with characteristic x, the situation may be represented by the list xi (i = 1,...,N) where xi is the characteristic of agent i. For example, if the characteristic is income, then xi is the income of agent i.
The Theil T index is defined as
and the Theil L index is defined as
where is the mean income:
Theil-L is an income-distribution's dis-entropy per person, measured with respect to maximum entropy (...which is achieved with complete equality).
(In an alternative interpretation of it, Theil-L is the natural-logarithm of the geometric-mean of the ratio: (mean income)/(income i), over all the incomes. The related Atkinson(1) is just 1 minus the geometric-mean of (income i)/(mean income),over the income distribution.)
Because a transfer between a larger income & a smaller one will change the smaller income's ratio more than it changes the larger income's ratio, the transfer-principle is satisfied by this index.
Official source
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