Not to exceed that minimum size would require a perfectly planned redistribution. Therefore, the Hoover index is the "non-stochastic" counterpart to the "stochastic" Theil index. In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions: . The Gini is the sum, over all income-ordered population-percentiles, of the shortfall, from equal-share, of the cumulative-income up to each population-percentile, with that summed shortfall divided by the greatest value that it could have, with complete inequality. In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. The normal distribution is the only distribution whose cumulants beyond the first two (i.e., other than the mean and variance) are zero.It is also the continuous distribution with the maximum entropy for a specified mean and variance. Do government programs support some income groups more than others? Table A1: Top fractiles income shares (excluding capital gains) in the U.S., 19132005", Striking it Richer: The Evolution of Top Incomes in the United States, "Inequality and Growth in a Panel of Countries", "Inequality, Growth and Poverty in the Era of Liberalization and Globalization", "Inequality of opportunity, inequality of income and economic growth", International Encyclopedia of the Social & Behavioral Sciences, Two Americas: One Rich, One Poor? According to the work "Inequality of Opportunity, Income Inequality, and Economic Growth," the relationship between income inequality and economic growth is mediated by the level of equality of opportunity, as identified by intergenerational mobility. The mode is the point of global maximum of the probability density function. In probability theory and statistics, the Gumbel distribution (also known as the type-I generalized extreme value distribution) is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions.. The Bayesian interpretation of probability can be seen as an extension of propositional logic that Income inequality metrics or income distribution metrics are used by social scientists to measure the distribution of income and economic inequality among the participants in a particular economy, such as that of a specific country or of the world in general. The precise shape of the inequality-growth curve obviously varies across countries depending upon their resource endowment, history, remaining levels of absolute poverty and available stock of social programs, as well as on the distribution of physical and human capital. The Atkinson index is defined in reference to a corresponding social welfare function, where mean income multiplied by one minus the Atkinson index gives the welfare equivalent equally distributed income. This affects the conclusions which can be drawn from a measured inequality. For each group the number of individuals (or households) per group A and the total income in that group E is specified. 4.97 Income distribution has always been a central concern of economic theory and economic policy. Keeping these points in mind helps to understand the problems caused by the improper use of inequality measures. The Galt score is a simple ratio of a company's CEO pay to the pay of that company's median worker. When applied to income distributions, the first Theil index (Theil-L) relates to how income is distributed to income earners, whereas the second Theil index (Theil-T) relates to how income-earners are distributed to incomes. In the physics of heat conduction , the folded normal distribution is a fundamental solution of the heat equation on the half space; it corresponds to having a perfect insulator on a hyperplane through the origin. The reason for its popularity is that it is easy to understand how to compute the Gini index as a ratio of two areas in Lorenz curve diagrams. Some special observations: For an 80:20 distribution (Pareto principle) the Theil index is 0.83. In other cases the groups are created based on income ranges which leads to having different numbers of individuals in the different groups. times fraction under equality), and the top 0.01% of US earners accounted for 6% of total wages (600 times fraction under equality). Despite its name, the first explicit analysis of the properties of the Cauchy distribution was published by the French In probability theory, the inverse Gaussian distribution (also known as the Wald distribution) is a two-parameter family of continuous probability distributions with support on (0,).. Its probability density function is given by (;,) = (())for x > 0, where > is the mean and > is the shape parameter.. This caused confusion. These statistics are easy to interpret and communicate, because they are relative (this population earns twice as much as this population), but, since they do not fall on an absolute scale, do not provide an absolute measure of inequality. In their study for the World Institute for Development Economics Research, Giovanni Andrea Cornia and Julius Court (2001) reach slightly different conclusions. Special case of distribution parametrization. qqplot(x) displays a quantile-quantile plot of the quantiles of the sample data x versus the theoretical quantile values from a normal distribution.If the distribution of x is normal, then the data plot appears linear.. qqplot plots each data point in x using plus sign ('+') markers and draws two reference lines that represent the theoretical distribution. Each paper writer passes a series of grammar and vocabulary tests before joining our team. In other words, it is the total of employees' income divided by the national income. Sometimes it is specified by only scale and shape and sometimes only by its shape parameter. In probability theory and statistics, the chi distribution is a continuous probability distribution.It is the distribution of the positive square root of the sum of squares of a set of independent random variables each following a standard normal distribution, or equivalently, the distribution of the Euclidean distance of the random variables from the origin. One form of income is the total amount of goods and services that a person receives, and thus there is not necessarily money or cash involved. For the Theil index also the term "Theil entropy" had been used. In many cases the inequality indices mentioned above are computed from such segment data without evaluating the inequalities within the segments. A weaker reduction of complexity is achieved if income distributions are described by shares of total income. A scale-free network is a network whose degree distribution follows a power law, at least asymptotically.That is, the fraction P(k) of nodes in the network having k connections to other nodes goes for large values of k as where is a parameter whose value is typically in the range < < (wherein the second moment (scale parameter) of is infinite but the first moment is finite), It follows that the Voigt profile will not have a moment-generating function either, but the characteristic function for the Cauchy distribution is well defined, as is the characteristic function for the normal distribution. Services like public health and education are also counted in. The Gini index is the most frequently used inequality index. The cumulative distribution function (CDF) can be written in terms of I, the regularized incomplete beta function.For t > 0, = = (,),where = +.Other values would be obtained by symmetry. Both very high egalitarianism and very high inequality cause slow growth. The probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set {,,, };; The probability distribution of the number Y = X 1 of failures before the first success, supported on the set {,,, }. The Atkinson index is for no value of highly sensitive to top incomes because of the common restriction that is nonnegative. Modern economists have also addressed this issue, but have been more concerned with the distribution of income across individuals and households. This distribution might be used to represent the distribution of the maximum level of a river in a particular year if there was a list of maximum Similarly, applying the Hoover index does not imply that allocation processes occur in a perfectly planned economy: the Hoover index yields the distance between the resource distribution in an observed system to the final stage of a planned "equalization" of resource distribution. What effect will inflation have on absolute measures? As for any resource distribution and with reference to information theory, "maximum entropy" occurs once income earners cannot be distinguished by their resources, i.e. The cumulative distribution function of the Gumbel distribution is (;,) = /.Standard Gumbel distribution. For 73:27 the Theil index and the Hoover index are identical: Both are 0.46. Clarify whether real or nominal income distributions should be used. In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables.Up to rescaling, it coincides with the chi distribution with two degrees of freedom.The distribution is named after Lord Rayleigh (/ r e l i /).. A Rayleigh distribution is often observed when the overall magnitude of a vector is related Some references give the shape parameter as =. On a global scale, the current carbon Palma ratio is noticeably higher than within any country, indicating an exceedingly severe inequality when individual emissions are considered outside territorial boundaries.[15]. Examples of such univariate distributions are: normal distributions, Poisson distributions, binomial distributions (with common success probability), negative binomial distributions (with common success probability), Neither does using the Theil index necessarily imply that a very low inequality (low redundancy, high entropy) is "good", because high entropy is associated with slow, weak and inefficient resource allocation processes. The probability density function (PDF) of the beta distribution, for 0 x 1, and shape parameters , > 0, is a power function of the variable x and of its reflection (1 x) as follows: (;,) = = () = (+) () = (,) ()where (z) is the gamma function.The beta function, , is a normalization constant to ensure that the total probability is 1. Because income distribution is generally positively skewed, mean is higher than median, so ratios to mean are lower than ratios to median. In probability theory and statistics, the Gumbel distribution (also known as the type-I generalized extreme value distribution) is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions.. It follows that the Voigt profile will not have a moment-generating function either, but the characteristic function for the Cauchy distribution is well defined, as is the characteristic function for the normal distribution. In real societies people can be distinguished by their different resources, with the resources being incomes. The differences between the Census and Internal Revenue Service Data can be seen most easily in the 95th percentile, present in both data sets. [29] According to Pak Hung Mo, income inequality has significant negative effect on the rate of GDP growth. For small , the quantile function has the useful asymptotic expansion = + ().. Properties. In probability theory, the multinomial distribution is a generalization of the binomial distribution.For example, it models the probability of counts for each side of a k-sided die rolled n times. has If it has a distribution from the same family of distributions as the original variables, that family of distributions is said to be closed under convolution.. Often expenditure or consumption (which is the same in an economic sense) is used to measure income. [13], The Palma ratio addresses the Gini index's over-sensitivity to changes in the middle of the distribution and insensitivity to changes at the top and bottom,[14] and therefore more accurately reflects income inequality's economic impacts on society as a whole. Applied to the analysis of income inequality within countries, "income" often stands for the taxed income per individual or per household. In probability theory, the multinomial distribution is a generalization of the binomial distribution.For example, it models the probability of counts for each side of a k-sided die rolled n times. This in a way solves the thorny issue of diverging moments in the stable distribution. When drawing conclusion from inequality measurements, consider how we should allocate the benefits of government spending? The expected value of a random variable with a finite In the physics of heat conduction , the folded normal distribution is a fundamental solution of the heat equation on the half space; it corresponds to having a perfect insulator on a hyperplane through the origin. Despite its name, the first explicit analysis of the properties of the Cauchy distribution was published by the French The standard Gumbel distribution is the case where = and = with cumulative distribution function = ()and probability density function = (+).In this case the mode is 0, the median is ( ()), the mean is (the EulerMascheroni constant), and the standard deviation is / The Theil index is an entropy measure. The Hoover index is the simplest of all inequality measures to calculate: It is the proportion of all income which would have to be redistributed to achieve a state of perfect equality. In probability theory, the inverse Gaussian distribution (also known as the Wald distribution) is a two-parameter family of continuous probability distributions with support on (0,).. Its probability density function is given by (;,) = (())for x > 0, where > is the mean and > is the shape parameter.. These consist of questionnaires with more than 200 questions. The modified half-normal distribution (MHN) is a three-parameter family of continuous probability distributions supported on the positive part of the real line. The World Bank uses the so-called "living standard measurement surveys"[1] to measure income. In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. The expected value of a random variable with a finite In the physics of heat conduction , the folded normal distribution is a fundamental solution of the heat equation on the half space; it corresponds to having a perfect insulator on a hyperplane through the origin. The long tail is the name for a long-known feature of some statistical distributions (such as Zipf, power laws, Pareto distributions and general Lvy distributions).In "long-tailed" distributions a high-frequency or high-amplitude population is followed by a low-frequency or low-amplitude population which gradually "tails off" asymptotically.The events at the far end of the tail have a This reflects that the increase in the Gini coefficient of the US in this time period is due to gains by upper income earners (relative to the median), rather than by losses by lower income earners (relative to the median). qqplot(x) displays a quantile-quantile plot of the quantiles of the sample data x versus the theoretical quantile values from a normal distribution.If the distribution of x is normal, then the data plot appears linear.. qqplot plots each data point in x using plus sign ('+') markers and draws two reference lines that represent the theoretical distribution. In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. This is particularly used to measure that fraction of income accruing to top earners top 10%, 1%, 0.1%, 0.01%, and also "top 100" earners or the like; in the US top 400 earners is 0.0002% of earners (2 in 1,000,000) to study concentration of income wealth condensation, or rather income condensation. High inequality yields high Theil redundancies. report several Lorenz-curve data-points (cumulative national income-shares up to several percentiles). In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and the class of compound Poissongamma distributions which have positive mass at zero, but are otherwise continuous. And households have a varying number of members. Evidence from a broad panel of recent academic studies shows that there is a nonlinear relation between income inequality and the rate of growth and investment. For n independent trials each of which leads to a success for exactly one of k categories, with each category having a given fixed success probability, the multinomial distribution gives The Gini coefficient, the Hoover index and the Theil index as well as the related welfare functions[27] can be computed together in a spreadsheet. In addition to a self-contained presentation of the Gini index, we give two equivalent ways to interpret this summary statistic: first in terms of the percentile level of the person who earns the average dollar, and second in terms of how the lower of two randomly chosen incomes compares, on average, to mean income.[3]. Cumulative distribution function. An important property of the Theil index which makes its application popular is its decomposability into the between-group and within-group component. The more "distinguishable" they are, the lower is the "actual entropy" of a system consisting of income and income earners. In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.It is a particular case of the gamma distribution.It is the continuous analogue of the geometric distribution, and it has the key In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables.Up to rescaling, it coincides with the chi distribution with two degrees of freedom.The distribution is named after Lord Rayleigh (/ r e l i /).. A Rayleigh distribution is often observed when the overall magnitude of a vector is related As a result, two very different distributions of income can have the same Gini index. The modified half-normal distribution (MHN) is a three-parameter family of continuous probability distributions supported on the positive part of the real line. A scale-free network is a network whose degree distribution follows a power law, at least asymptotically.That is, the fraction P(k) of nodes in the network having k connections to other nodes goes for large values of k as where is a parameter whose value is typically in the range < < (wherein the second moment (scale parameter) of is infinite but the first moment is finite), In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions.The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distribution. For more details see the help file for the "incomeInequality" data in the Ecdat package available from the Comprehensive R Archive Network (CRAN; see r-project.org). In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and the class of compound Poissongamma distributions which have positive mass at zero, but are otherwise continuous. After a number of recently developed internationally comparable measures of intergenerational mobility they confirmed that the negative impact of income inequality on growth increases as intergenerational mobility decreases. 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. While different theories may try to explain how income inequality comes about, income inequality metrics simply provide a system of measurement used to determine the dispersion of incomes. A Theil index of 0 indicates perfect equality. Considering the inequalities in economically well developed countries, public policy should target an 'efficient inequality range'. If the two countries' Lorenz curves cross, then it's a near-certainty that, on the poor side of the crossing-point, country A has a higher Lorenz curve, meaning that, for any poorest X% of the population up to the crossing-point, country A's population has a greater national-income-share. A Theil index of 0.5 characterizes systems which are close to a 74:26 distribution. Normative approach has been discussed by Charles Blackorby, Walter Bossert and David Donaldson in the work "Income Inequality Measurement: The Normative Approach". It has the advantages of being mathematically tractable and its square is subgroup decomposable, but it is not bounded from above. (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. In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one.. [Note 5] The welfare functions serve as alternatives to the median income. For example, the attached graph shows that in the period 19672003, US income ratio between median and 10th and 20th percentile did not change significantly, while the ratio between the median and 80th, 90th, and 95th percentile increased. R is a shift parameter, [,], called the skewness parameter, is a measure of asymmetry.Notice that in this context the usual skewness is not well defined, as for < the distribution does not admit 2nd or higher moments, and the usual skewness definition is the 3rd central moment.. This information reduction of course is the goal of computing inequality measures, as it reduces complexity. The 20:20 or 20/20 ratio compares how much richer the top 20% of populations are to the bottom 20% of a given population. Classical economists such as Adam Smith, Thomas Malthus and David Ricardo were mainly concerned with factor income distribution, that is, the distribution of income between the main factors of production, land, labour and capital. But this does not necessarily imply that a very high inequality is "good", because very low entropies also can lead to explosive compensation processes. A Theil index of 1 indicates that the distributional entropy of the system under investigation is almost similar to a system with an 82:18 distribution. (If the inequality within the segments is known, the total inequality can be determined by those inequality metrics which have the property of being "decomposable".). The truncated normal distribution, half-normal distribution, and square-root of the Gamma distribution are special cases of the MHN distribution. Cumulative distribution function. The distribution is called "folded" because probability mass to the left of x = 0 is folded over by taking the absolute value. A company which pays its CEO many times more than its median employee will have a high Galt score. Sometimes it is specified by only scale and shape and sometimes only by its shape parameter. The findings imply that excluding intergenerational mobility leads to misspecification, which explains why the empirical literature on income inequality and growth has been so inconclusive. If the two countries' Lorenz curves don't cross, then country A's Lorenz curve is everywhere higher than that of country B, meaning that, for any X from 0 to 100, the poorest X% of the population have a greater national-income-share in country A than in country B. The Theil index indicates the distributional redundancy of a system, within which incomes are assigned to income earners in a stochastic process. 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