Theil 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.
[3] 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.
"[4] 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.
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).
if the situation is characterized by a continuous distribution function f(k) (supported from 0 to infinity) where f(k) dk is the fraction of the population with income k to k + dk, then the Theil index is: where the mean is: Theil indices for some common continuous probability distributions are given in the table below: If everyone has the same income, then TT equals 0.
The Theil index measures an entropic "distance" the population is away from the egalitarian state of everyone having the same income.
The numerical result is in terms of negative entropy so that a higher number indicates more order that is further away from the complete equality.
The Theil index is derived from Shannon's measure of information entropy
In information theory, physics, and the Theil index, the general form of entropy is When looking at the distribution of income in a population,
If the Theil index is used with x=population/species, it is a measure of inequality of population among a set of species, or "bio-isolation" as opposed to "wealth isolation".
The Theil index measures what is called redundancy in information theory.
A high Theil index indicates the total income is not distributed evenly among individuals in the same way an uncompressed text file does not have a similar number of byte locations assigned to the available unique byte characters.
), both of which allow one to decompose inequality into the part that is due to inequality within areas (e.g. urban, rural) and the part that is due to differences between areas (e.g. the rural-urban income gap).
The decomposition of the Theil index which identifies the share attributable to the between-region component becomes a helpful tool for the positive analysis of regional inequality as it suggests the relative importance of spatial dimension of inequality.
Indexes of inequality in the generalized entropy (GE) family are more sensitive to differences in income shares among the poor or among the rich depending on a parameter that defines the GE index.
The smaller the parameter value for GE, the more sensitive it is to differences at the bottom of the distribution.
[9] The decomposability is a property of the Theil index which the more popular Gini coefficient does not offer.
The Gini coefficient is more intuitive to many people since it is based on the Lorenz curve.
In addition to multitude of economic applications, the Theil index has been applied to assess performance of irrigation systems[10] and distribution of software metrics.
