Theil index - AbsoluteAstronomy.com

The Theil index is a statistic used to measure economic inequality

Economic inequality

Economic inequality comprises all disparities in the distribution of economic assets and income. The term typically refers to inequality among individuals and groups within a society, but can also refer to inequality among countries. The issue of economic inequality is related to the ideas of...

. It has also been used to measure the lack of racial diversity. The basic Theil index T_T is the same as redundancy in information theory

Redundancy (information theory)

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. Informally, it is the amount of wasted "space" used to transmit certain data...

which is the maximum possible entropy of the data minus the observed entropy. It is a special case of the generalized entropy index

Generalized entropy index

The generalized entropy index is a general formula for measuring redundancy in data. The redundancy can be viewed as inequality, lack of diversity, non-randomness, compressibility, or segregation in the data. The primary use is for income inequality...

. It can be viewed as a measure of redundancy, lack of diversity, isolation, segregation, inequality, non-randomness, and compressibility. It was proposed by econometrician

Econometrics

Econometrics has been defined as "the application of mathematics and statistical methods to economic data" and described as the branch of economics "that aims to give empirical content to economic relations." More precisely, it is "the quantitative analysis of actual economic phenomena based on...

Henri Theil

Henri Theil was a Dutch econometrician.He graduated from the University of Amsterdam. He was the successor of Jan Tinbergen at the Erasmus University Rotterdam. Later he taught in Chicago and at the University of Florida. He is most famous for his invention of 2-stage least squares...

, a successor of Jan Tinbergen

Jan Tinbergen

Jan Tinbergen , was a Dutch economist. He was awarded the first Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel in 1969, which he shared with Ragnar Frisch for having developed and applied dynamic models for the analysis of economic processes...

at the Erasmus University Rotterdam.

Formulas

The basic Theil index, which has higher resolution for changes to higher incomes, is

where

is income/person. When

is inverted to be people/income, or if changes in lower incomes are more important, a different formula is used that is derivable from

is also known as the MLD (mean log deviation) because it gives the standard deviation of

. Sometimes the average of

and

is used, which has the advantage of being "symmetric" like the Gini, Hoover, and Coulter indices. "Symmetric" means it gives the same result for x as it does for 1/x:

For these equations,

is the income of the

th person or subgroup,

is the mean income of the persons or subgroups, and

is the population or number of subgroups.

If everyone has the same income, the indices give 0 which, counter-intuitively, is when the population's income has maximum disorder. If one person has all the income, then T_T gives the result

, which is maximum order. Dividing T_T by

can normalize the equation to range from 0 to 1.

The indices measure an entropic "distance" the population is away from the "ideal" 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 "ideal" of maximum disorder. Formulating the index to represent negative entropy instead of entropy allows it to be a measure of inequality rather than equality.

If

applies to the distribution of income in people, then

can be used to get the same numerical result for the distribution of people in income.

The two Theil indices

and

are special cases of the generalized entropy index

Generalized entropy index

with

and

. The Atkinson index

Atkinson index

The Atkinson index is a measure of income inequality developed by British economist Anthony Barnes Atkinson...

with

is a transformation of

by A=1-e^-T. Likewise, the Atkinson index with

is a transformation of

Derivation from Entropy

The Theil index is derived from Shannon's measure of information entropy

Information entropy

In information theory, entropy is a measure of the uncertainty associated with a random variable. In this context, the term usually refers to the Shannon entropy, which quantifies the expected value of the information contained in a message, usually in units such as bits...

(S), where entropy is a measure of randomness in a given set of information. In information theory, physics, and the Theil index, the general form of entropy is

where p_i is the probability of finding member i from a random sample of the population. In physics, k is Boltzmann's constant. In information theory k=1 if it is in terms of bits and the log base is 2. Physics and the Theil index have chosen the natural logarithm as the logarithmic base. When p_i is chosen to be income per person (x_i), it needs to be normalized by dividing by the total population income, N*avg(x). This gives the observed entropy of a Theil population to be:

The Theil index is T_T = S_max - S_Theil where the theoretical maximum entropy S_max is when all incomes are equal, i.e. each x_i = average x_i = a constant. This is substituted into S_Theil to give S_max = ln(N) for T_T, a constant determined solely by the population. So the Theil index gives a value in terms of an entropy that measures how far S_Theil is away from the "ideal" S_max. The index is a "negative entropy" in the sense that it gets smaller as the disorder gets larger, so it is a measure of order rather than disorder.

When x is in units of population/species,

is a measure of biodiversity and is called the Shannon index

Shannon index

The Shannon index, sometimes referred to as the Shannon-Wiener Index or the Shannon-Weaver Index, is one of several diversity indices used to measure diversity in categorical data. It is simply the Information entropy of the distribution, treating species as symbols and their relative population...

. If the Thiel 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

Redundancy (information theory)

in information theory. It is the left over "information space" that was not utilized to convey information, which reduces the effectiveness of the price signal

Price signal

A price signal is a message sent to consumers and producers in the form of a price charged for a commodity; this is seen as indicating a signal for producers to increase supplies and/or consumers to reduce demand.- Free price system :...

. The Theil index is a measure of the redundancy of income (or other measure of wealth) in some individuals. Redundancy in some individuals implies scarcity in others. 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.

Notation	Information Theory	Theil Index T_T
N	number of unique characters	number of individuals
i	a particular character	a particular individual
x_i	character_i count	income of individual_i
N*avg(x)	total characters in document	total income in population
T_T	unused information space	unused potential in price mechanism
	data compression	progressive tax

Application of the Theil index

Theil's measure can be converted by the operation

into one of the indexes of Anthony Barnes Atkinson

Anthony Barnes Atkinson

Sir Anthony Barnes "Tony" Atkinson, FBA, is a British economist and has been a Senior Research Fellow of Nuffield College, Oxford since 2005.-Career:Atkinson served as Warden of Nuffield College from 1994 to 2005...

, where

may or may not be used to introduce an inequality aversion factor into the formula, with

being the default. The result of the conversion also has been called normalized Theil index.

James E. Foster used such a measure to replace the Gini coefficient in Amartya Sen

Amartya Sen

Amartya Sen, CH is an Indian economist who was awarded the 1998 Nobel Prize in Economic Sciences for his contributions to welfare economics and social choice theory, and for his interest in the problems of society's poorest members...

'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

, if the conversion is included into the computation of the average per capita welfare function

Social welfare function

In economics, a social welfare function is a real-valued function that ranks conceivable social states from lowest to highest. Inputs of the function include any variables considered to affect the economic welfare of a society...

Using the "Theil-L" index

(see below) for

in that formula yields results similar to using the Atkinson index

Atkinson index

The Atkinson index is a measure of income inequality developed by British economist Anthony Barnes Atkinson...

for computing the welfare function.

What does U mean ?

If U = 1 Then the "Naive" (NF1) Method is as good as the current Forecast Method

If U < 1 Then the Forecasting Method is better than the NF1 Method

If U > 1 Then the NF1 Method is better than the Forecasting Method. There is not need to waste time applying anymore Forecasting Techniques.

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 for the Theil-T index the population is divided into

certain subgroups and

is the income share of group

is the Theil-T index for that subgroup, and

is the average income in group

, then the Theil index is

The formula for the Theil-L index is:

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).

The decomposition of the overall 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.

The decomposability is a property of the Theil index which the more popular Gini coefficient

Gini coefficient

The Gini coefficient is a measure of statistical dispersion developed by the Italian statistician and sociologist Corrado Gini and published in his 1912 paper "Variability and Mutability" ....

does not offer. The Gini coefficient is more intuitive to many people since it is based on the Lorenz curve

Lorenz curve

In economics, the Lorenz curve is a graphical representation of the cumulative distribution function of the empirical probability distribution of wealth; it is a graph showing the proportion of the distribution assumed by the bottom y% of the values...

. However, it is not easily decomposable like the Theil.

Applications

In addition to multitude of economic applications, the Theil index has been applied to assess performance of irrigation

Irrigation

Irrigation may be defined as the science of artificial application of water to the land or soil. It is used to assist in the growing of agricultural crops, maintenance of landscapes, and revegetation of disturbed soils in dry areas and during periods of inadequate rainfall...

systems and distribution of software metrics.

External links

Software:
- Free Online Calculator computes the Gini Coefficient, plots the Lorenz curve, and computes many other measures of concentration for any dataset
- Free Calculator: Online and downloadable scripts (Python
  Python (programming language)
  Python is a general-purpose, high-level programming language whose design philosophy emphasizes code readability. Python claims to "[combine] remarkable power with very clear syntax", and its standard library is large and comprehensive...
  
  and Lua
  Lua programming language
  Lua is a lightweight multi-paradigm programming language designed as a scripting language with extensible semantics as a primary goal. Lua has a relatively simple C API compared to other scripting languages.- History :...
  
  ) for Atkinson, Gini, and Hoover inequalities
- Users of the R data analysis software can install the "ineq" package which allows for computation of a variety of inequality indices including Gini, Atkinson, Theil.
- A MATLAB Inequality Package, including code for computing Gini, Atkinson, Theil indexes and for plotting the Lorenz Curve. Many examples are available.

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.