Theil Entropy Research Articles

EQUALITY MEASURES AND FACILITY LOCATION

ABSTRACT A review of the literature indicates that fairly simplistic measures of equality in travel distances are normally used in facility location studies. Two popular criteria, those of minimizing the maximal distance and the range, are known to be sensitive to the extreme values (locations of users) of the problem. This paper recommends that distributional equality should be used instead in most locational studies. Using a simple spatial setting, the paper offers a comparative analysis of equality curves based on the following measures: Gini coefficient, mean deviation, Hoover's concentration index, variance (standard deviation), and Theil's entropy index.

Equality measures and facility location

A review of the literature indicates that fairly simplistic measures of equality in travel distances are normally used in facility location studies. Two popular criteria, those of minimizing the maximal distance and the range, are known to be sensitive to the extreme values (locations of users) of the problem. This paper recommends that distributional equality should be used instead in most locational studies. Using a simple spatial setting, the paper offers a comparative analysis of equality curves based on the following measures: Gini coefficient, mean deviation, Hoover's concentration index, variance (standard deviation), and Theil's entropy index.

Entropy Measures of Inequality*

Although sociologists have long been interested in the analysis of inequality, there have been relatively few systematic attempts to measure it. Most measurement has utilized popular existing measures such as the Gini coefficient, which unfortunately is relatively difficult to compute and interpret. This article presents a new measure of inequality (B) that is based upon the concept of entropy. The B measure is used for categorical data and complements Theil's (T) entropy measure for continuous data. The B measure is illustrated for both income and wealth data and successfully meets several criteria for evaluating measures of inequality. These include the criteria of scale invariance, sensitivity to transfers, and adequate upper bounds. A further advantage of sociological entropy measures of inequality is that they facilitate interdisciplinary comparison of work on inequality between sociology and other disciplines (such as economics) which use entropy measures of inequality.

Decomposition of inequality measures and a comparative analysis

The main objectives of the paper are to decompose Atkinson's measure into within-group and between-group inequality and to compare it with decompositions of inequality measures, like the Gini coefficient, variance of logarithms, Theil's entropy index and the square of the coefficient of variation. Atkinson's inequality index is given an interpretation of mean order β and then it is decomposed for population subgroups. Various other measures are considered for decompositions and a variety of decomposition schemes for Gini coefficient and other measures in the literature is reviewed and commented upon. Empirical analysis using U.S.A. and U.K. data are performed on the basis of income distribution data classified by family size. For various decomposition schemes, contribution to between size-group and within size-group inequalities are compared in percentage terms. Comparisons are also made over time for the years 1964 and 1974 both for U.S.A. and U.K. and the result indicates that within size-group income inequality has declined while between size group and total income inequality shows an increase. Three important properties of Atkinson's inequality index are proved in the appendix.

Regression analysis of inequality measures applied to interval scales

Suppose it was wished to determine whether inequality of occupational status in the US had increased or declined since the 1950s. A straightforward approach to answering this question would be to use Duncan 's (1961) socioeconomic index (SEI) as the measure of occupational status, and to apply the Gini index of inequality to data collected at several points in time. Peter Blau (1977, p. 211) did just that for samples of US men of selected ages in 1952, 1962 and 1972. He reported Gini values of 0.353, 0.330 and 0.318 for the three pe r iods a clear decline in inequality over time. There is a fundamental problem with this approach, however. Under the most generous interpretation, the SEI is an interval rather than a ratio scale. Thus, unlike dollar income or number of children, the SEI scale does not possess a theoretically fixed zero point. It would be quite undesirable, then, if inferences about the distribution of occupational status were sensitive to the location of the arbitrary zero point. To put it another way, whatever conclusions are drawn about interval scales should be invariant to the addition of an arbitrary constant to each individual's score, since this is equivalent to changing the zero point. Unfortunately, the Gini index is quite sensitive to changes in the zero point (Allison, 1978). In fact, any scale-invariant measure of inequality will have an arbitrary value if applied to a variable with an arbitrary zero point. In addition to the Gini index, scale-invariant measures of inequality include the coefficient of variation, the relative mean deviation, the standard deviation of the logarithms, and Theil's entropy measure. All these measures possess the desirable property that if each sample member 's score is multiplied by a constant, the level of inequality remains unchanged [ 1]. On the other hand, adding a constant to each individual's score has a marked effect on all of these measures. This phenomenon can be specified more precisely for three of these measures: the Gini index, the coefficient of variation, and the relative mean deviation (a special case of the index of dissimilarity). The coefficient of

Industrial Decline, Regional Policy and the Urban—Rural Manufacturing Shift in the United Kingdom

The paper reports results of Theil entropy index, shift-share, and regression analyses of county variations in the change of manufacturing employment in the United Kingdom between 1971 and 1976. The research identified a very marked and consistent urban–rural shift in the relative distribution of manufacturing employment during this period, associated primarily with nonstructural influences. Regression tests of specific hypotheses derived from previous work, concerning the possible impact of government regional policy incentives, residential space preferences, and female labour availability (the ‘restructuring hypothesis’), failed to add significantly to the statistical explanation achieved. The paper concludes with a brief discussion of possible implications for UK regional policy in the 1980s.

The Class of Additively Decomposable Inequality Measures

This paper considers a wide class of inequality indices and identifies those which are additively decomposable. The sub-class of mean independent, additively decomposable measures turns out to be a single parameter family which includes the squared coefficient of variation and the two Theil's entropy formulas.

An Analysis of Some Properties of Alternative Measures of Income Inequality Based on the Gamma Distribution Function

Abstract The Gini, Theil entropy, and Pietra measures of inequality associated with the gamma distribution function are expressed in terms of the parameters defining the gamma distribution. Method of moments (MME) and maximum likelihood estimators (MLE) of these measures are obtained along with expressions for the asymptotic standard errors of the MLE measures. A table is presented that facilitates the calculation of MLE of inequality measures and the associated asymptotic standard errors by expressing each as a function of the ratio of the arithmetic mean and geometric mean. This table also facilitates the calculation of MME estimates of inequality measures. The results of a Monte Carlo study are used to compare the performance of the MME and MLE for data generated from a population characterized by a gamma distribution and to consider questions of statistical inference and requisite sample size.

An Analysis of Some Properties of Alternative Measures of Income Inequality Based on the Gamma Distribution Function

Abstract The Gini, Theil entropy, and Pietra measures of inequality associated with the gamma distribution function are expressed in terms of the parameters defining the gamma distribution. Method of moments (MME) and maximum likelihood estimators (MLE) of these measures are obtained along with expressions for the asymptotic standard errors of the MLE measures. A table is presented that facilitates the calculation of MLE of inequality measures and the associated asymptotic standard errors by expressing each as a function of the ratio of the arithmetic mean and geometric mean. This table also facilitates the calculation of MME estimates of inequality measures. The results of a Monte Carlo study are used to compare the performance of the MME and MLE for data generated from a population characterized by a gamma distribution and to consider questions of statistical inference and requisite sample size.

Within‐Set and Between‐Set Income Inequalities in a Developing Country

The relation between economic growth and income distribution has been the subject of a growing body of empirical studies ever since the initial discussion by Kuznets, who argued that the income distribution in the early stages of growth in developing areas would most likely move in the direction of greater inequality. While there is substantial support for this hypothesis of relative inequality increase, recent cross‐country evidence does not corroborate a stronger hypothesis of a decline in the absolute income level of the poorer groups. Nevertheless, cross‐sectional analyses may be fraught with misleading generalisations, and the validity of the hypothesis will have to be assessed by the secular experience of individual countries. Within this focus, one country which does not appear to support the Kuznets thesis is Puerto Rico, where recent studies point to a movement toward greater income equality over the 1949–69 period. This paper, which covers the same two‐decade interval, attempts to throw further light upon the overall question by employing, in addition to the well‐known Lorenz/Gini measure, Theil's entropy index. The latter coefficient permits decomposition of given sets by quantifying between‐set and within‐set inequalities.