NationMaster - Encyclopedia: Theil index

The Theil index^[1], derived by econometrician Henri Theil, is a statistic used to measure economic inequality. Econometrics is concerned with the tasks of developing and applying quantitative or statistical methods to the study and elucidation of economic principles. ... Henri Theil (13 October 1924 Amsterdam, 2000) was a a Dutch econometrician. ... Differences in national income equality around the world as measured by the national Gini coefficient. ...

1 Mathematics
2 Decomposability
3 Application of the Theil index
4 Pareto principle
5 Theil index and Hoover index
6 References

Mathematics

The formula is

where $x i$ is the income of the $i$ th person, $overline{x}= frac{1}{N} sum_{i=1}^N x_i$ is the mean income, and $N$ is the number of people. The first term inside the sum can be considered the individual's share of aggregate income, and the second term is that person's income relative to the mean. If everyone has the same (i.e., mean) income, then the index is 0. If one person has all the income, then the index is ln&nbspN}.

The Theil index is derived from Shannon's measure of information entropy. Letting T be the Theil index and S be Shannon's information entropy measure, Claude Shannon Claude Elwood Shannon (April 30, 1916 â€“ February 24, 2001), an American electrical engineer and mathematician, has been called the father of information theory,[1] and was the founder of practical digital circuit design theory. ... Claude Shannon In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. ...

$T=ln(N)-S. ,$

Shannon derived his entropy measure in terms of the probability of an event occurring. This can be interpreted in the Theil index as the probability a dollar drawn at random from the population came from a specific individual. This is the same as the first term, the individual's share of aggregate income. Probability is the likelihood that something is the case or will happen. ...

With reference to information theory^[2], Theil's measure is a redundancy rather than an entropy. The redundancy of a system at a given time is the difference between its maximum entropy and its present entropy at that time.^[3]

Decomposability

One of the advantages of the Theil index is that it is a weighted average of inequality within subgroups, plus inequality among those subgroups. For example, inequality within the United States is the average inequality within each state, weighted by state income, plus the inequality among states.

If the population is divided into $m$ certain subgroups and $s k$ is the income share of group $k$ , $T k$ is the Theil index for that subgroup, and $overline{x}_k$ is the average income in group $k$ , then the Theil index is

$T = sum_{k=1}^m s_k T_k + sum_{k=1}^m s_k ln{frac{overline{x}_k}{overline{x}}}.$

Another, more popular, measure of inequality is the Gini coefficient. The Gini coefficient is more intuitive to many people since it is based on the Lorenz curve. However, it is not easily decomposable like the Theil. Graphical representation of the Gini coefficient The Gini coefficient is a measure of inequality of income distribution or inequality of wealth distribution. ... The Lorenz curve is a graphical representation of the cumulative distribution function of a probability distribution; it is a graph showing the proportion of the distribution assumed by the bottom y% of the values. ...

Application of the Theil index

Theil's index takes an equal distribution for reference which is similar to distributions in statistical physics. An index for an actual system is an actual redundancy, that is, the difference between maximum entropy and actual entropy of that system.

Theil's measure can be converted^[3] into one of the indexes of Anthony Barnes Atkinson. The result of the conversion also is called normalized Theil index^[4]. James E. Foster^[5] used such a measure to replace the Gini coefficient in Amartya Sen's welfare function W=f(income,inequality). The income e.g. is the average income for individuals in a group of income earners. Thus, Foster's welfare function can be computed directly from the Theil index T, if the conversion is included into the computation of the average per capita welfare function: Sir Anthony Barnes Atkinson (Tony Atkinson) is a British economist and currently Warden of Nuffield College, Oxford since August 1994. ... This article does not cite any references or sources. ... A social welfare function, in welfare economics, is a function which gives a measure of the material welfare of society, given a number of economic variables as inputs. ...

$W = overlinetext{income} times {e^{-T}}.,$

Note: This image is not the Theil Index in each area of the United States, but of contributions to the US Theil Index by each area (the Theil Index is always positive, individual contributions to the Theil Index may be negative or positive). Image File history File links Size of this preview: 776 Ã— 600 pixelsFull resolution (3300 Ã— 2550 pixel, file size: 279 KB, MIME type: image/png) File historyClick on a date/time to view the file as it appeared at that time. ...

Pareto principle

For ressource distributions described by only two quantiles, the Theil index is 0 for 50:50 distributions and reaches 1 at 82:18^[6], which is very close to a distribution often referred to as "Pareto Principle". Higher inequities yield Theil indices above 1. The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution found in a large number of real-world situations. ...

Theil index and Hoover index

For the income distributions provided by the The World Income Inequality Database (2007-05) the difference between their symmetrized Theil indices and their Hoover indices are plotted over their respective Gini indices. The difference illustrates the impact of the different inequalities on the information generated by them. Negative values occur for Theil indices, which are smaller than the respective Hoover indices.

For the income distributions provided by the The World Income Inequality Database (2007-05)^[7] the difference between their symmetrized Theil indices and their Hoover indices are plotted over their respective Gini indices. The difference illustrates the impact of the different inequalities on the information generated by them. Negative values occur for Theil indices, which are smaller than the respective Hoover indices.

A comparison of the Hoover index (also called Robin Hood index) and the Theil index shows the meaning of both indices: Image File history File links Metadata No higher resolution available. ... Image File history File links Metadata No higher resolution available. ... The Robin Hood index is a measure of income inequality. ...

For the Hoover index, the relative deviations in each quantile are summed up. Each deviation is weighted by its own sign (+1 or −1). Thus, the Hoover index is the most simple inequality measure. It has no normative foundations and does not refer to any models from physics or information theory.
For the symmetrized Theil index, the relative deviations in each quantile are summed up as well. But each deviation is weighted by its relative information weight. Thus, the Theil index is an indicator not only for the plain relative inequality, it also attempts to indicate how much attention inequality can get.

The following formulas illustrate that difference in the categories symmetry and percevability. For the formulas, a notation^[8] is used, where the amount $N$ of quantiles only appears as upper border of summations. Thus, inequities can be computed for quantiles with different widths $A i$ . For example, $E i$ could be the income in the quantile #i and $A i$ could be the amount (absolute or relative) of earners in the quantile #i. $E total$ then would be the sum of incomes of all $N$ quantiles and $A total$ would be the sum of the income earners in all $N$ quantiles. For evaluation of sums in closed form see evaluating sums. ...

Computation of the (asymmetric) Theil index T ^[9]:

$T = ln{frac{{A}_text{total}}{{E}_text{total}}} - frac{sum_{i=1}^N {{E}_i} ln{frac{{A}_i}{{E}_i}}}{{E}_text{total}}.$

With normalized data, $E' i = E i / E total$ and $A' i = A i / A total$ would apply. This would simplify the formula:
$color{Gray} T = 0 - frac{sum_{i=1}^N {{E}'_i} ln{frac{{A}'_i}{{E}'_i}}}{1} = sum_{i=1}^N {{E}'_i} ln{frac{{E}'_i}{{A}'_i}}$

Computation of the symmetrized Theil index $T s$ :

$T_s = frac{1}{2} left( ln{frac{{A}_text{total}}{{E}_text{total}}} - frac{sum_{i=1}^N {{E}_i} ln{frac{{A}_i}{{E}_i}}}{{E}_text{total}} + ln{frac{{E}_text{total}}{{A}_text{total}}} - frac{sum_{i=1}^N {{A}_i} ln{frac{{E}_i}{{A}_i}}}{{A}_text{total}} right).$

This leads to:

For comparison, the Hoover index $H$ :

$H = {frac{1}{2}} sum_{i=1}^N color{Blue} color{Black} left| {frac{{E}_i}{{E}_text{total}}} - {frac{{A}_i}{{A}_text{total}}} right|.$

References

^ Introduction to the Theil index from the University of Texas
^ ISO/IEC DIS 2382-16:1996 Information theory
^ ^a ^b http://www.poorcity.richcity.org (Redundancy, Entropy and Inequality Measures)
^ Juana Domínguez-Domínguez, José Javier Núñez-Velázquez: The Evolution of Economic Inequality in the EU Countries During the Nineties, 2005
^ James E. Foster and Amartya Sen, 1996, On Economic Inequality, expanded edition with annexe, ISBN 0-19-828193-5
^ 82.4% of the people own 17.6% of all ressources and 17.6% own 82.4% of all ressources. For computation see also http://www.poorcity.richcity.org/calculator/?quantiles=82.4,17.6|17.6,82.4
^ http://www.wider.unu.edu/wiid/wiid.htm
^ The notation using E and A follows the notation of a small calculus published by Lionnel Maugis: Inequality Measures in Mathematical Programming for the Air Traffic Flow Management Problem with En-Route Capacities (für IFORS 96), 1996
^ (1) The first part of the formula is the maximum entropy of the E-A-system. The second part (after the minus symbol) is the real entropy of the E-A-system at a certain time. Such a difference is called redundancy (ISO/IEC DIS 2382-16, information theory).
(2) This version of Theil's formula allows to process quantiles with different widths $A i$ . $N$ only serves as summation index.
(3) Besides mathematical comparison of this formula to the formulas found in many calculuses, you can compare the results 1A and 1B yielded by this formula with the examples 1A and 1B given in The Theoretical Basics of Popular Inequality Measures (Travis Hale, University of Texas Inequality Project, 2003).

Categories: Econometrics | Information theory | Economic indicators | Index numbers | Income distribution | Welfare economics Not to be confused with information technology, information science, or informatics. ... This article does not cite any references or sources. ... Redundancy in information theory is the number of bits used to transmit a message minus the number of bits of actual information in the message. ... Not to be confused with information technology, information science, or informatics. ...

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