The Measurement and Decomposition of Multi-Dimensional Inequality

Maasoumi, Esfandiar

doi:10.2307/1912849

“…As proved in Maasoumi (1986), the GEM is additively decomposable in the sense of (8) to (10). The values of this index range from 0 to infinity; its components are shown in Table 2, wheres is the arithmetic mean of the functions s i over all N individuals, ands g is the arithmetic mean of s i over the individuals in subgroup g.…”

Section: Polarization Via Additive Inequality Decompositionmentioning

confidence: 87%

“…As an index of multivariate inequality, Maasoumi (1986) proposed the following generalized entropy measure (henceforth, GEM):…”

Section: Polarization Via Additive Inequality Decompositionmentioning

confidence: 99%

“…For the particular functions f , h and s i , chosen by Maasoumi (1986) and Tsui (1995), and following the approach of Lasso-de-la Vega and Urrutia (2003), the similarity measure A in (21) can be multiplicatively decomposed into…”

Section: Multiplicative Decompositionmentioning

confidence: 99%

“…Some of the inequality measures considered above are so flexible to allow for different kinds of association between attributes; they are the GEM and AM measures. The aggregative function s i , introduced by Maasoumi (1986), is, in fact, based on the parameter β, which expresses the degree of substitution between attributes, such that β = (1/σ) − 1, where σ is a constant elasticity of substitution. So, if two attributes are substitutes, σ tends to infinity and, correspondingly, β → −1.…”

Section: Interaction Between Attributesmentioning

confidence: 99%

“…In the measurement of economic inequality and poverty, several authors have pointed out that attributes beyond income should be included in the analysis. Consequently, they have introduced various multi-attribute measures of inequality and poverty; see Atkinson and Bourguignon (1982), Kolm (1977), Maasoumi (1986), Bourguignon and Chakravarty (2003), Tsui (1995Tsui ( , 1999.…”

Section: Introductionmentioning

confidence: 99%

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Multivariate indices of polarization are constructed to measure effects of non-income attributes like wealth and education. Adopting the view of Esteban and Ray (1994), polarization is considered as the presence of groups which are internally homogeneous, externally heterogeneous, and of similar size. We propose a class of polarization indices which is built from measures of relative groups size and from decomposable indices of socio-economic inequality. For the latter, we employ the special inequality indices of Maasoumi (1986), Tsui (1995Tsui ( , 1999 and Koshevoy and Mosler (1997). Then, postulates for multidimensional polarization measurement are stated and discussed. The approach is illustrated by a numerical example.

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In this paper we consider the use of bootstrap methods to compute interval estimates and perform hypothesis tests for decomposable measures of economic inequality. The bootstrap potentially represents a significant gain over available asymptotic intervals because it provides an easily implemented solution to the Behrens-Fisher problem. Two applications of this approach, using the PSID (for the study of taxation) and the XLSY (for the study of youth inequality), to the Gini coefficient and Theil's entropy measures of inequality, are provided. The results suggest that (i) statistical inference is essential even when large samples are available, and (ii) the bootstrap appears to perform well in this setting. 'Cowell(1989a) is a notable esception. 2See, for example, Gastwirth (19i4), Gastwirth et al. (1986) and Cowell (1989a). (1994) provides a thorough review and some examples of use of asymptotic results. 1 Maasoumi small sample properties of these intervals are not known. Further, all the decomposable inequality measures used in the literature are bounded (e.g. the Gini coefficient lies in the [0, l] ' t m erval), so that application of standard asymptotic results may lead to estimated intervals that extend beyond the theoretical bounds of a particular measure (e.g. a negative lower bound for Gini). An alternative method for computing probability intervals is to bootstrap. The bootstrap provides interval estimates drawn from the small sample distribution. These interval estimates have been shown to be superior to asymptotic intervals both the-3 oretically and in a variety of applications.. Bootstrap intervals are computationally inexpensive and easy to calculate, the same method applies to all the inequality measures used in the literature, and the bootstrap method automatically takes into account any bounds that, apply to a particular measure. Further, since bootstrap intervals computed using the percentile met.hod have a clear Bayesian interpretation, they provide a straightforward solution to the Behrens-Fisher problem of comparing means from two distributions (see section 3).

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Statistical Inference via Bootstrapping for Measures of Inequality

MILLS¹,

ZANDVAKILI²

1997

J. Appl. Econ.

16

0

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In this paper we consider the use of bootstrap methods to compute interval estimates and perform hypothesis tests for decomposable measures of economic inequality. The bootstrap potentially represents a significant gain over available asymptotic intervals because it provides an easily implemented solution to the Behrens-Fisher problem. Two applications of this approach, using the PSID (for the study of taxation) and the XLSY (for the study of youth inequality), to the Gini coefficient and Theil's entropy measures of inequality, are provided. The results suggest that (i) statistical inference is essential even when large samples are available, and (ii) the bootstrap appears to perform well in this setting. 'Cowell(1989a) is a notable esception. 2See, for example, Gastwirth (19i4), Gastwirth et al. (1986) and Cowell (1989a). (1994) provides a thorough review and some examples of use of asymptotic results. 1 Maasoumi small sample properties of these intervals are not known. Further, all the decomposable inequality measures used in the literature are bounded (e.g. the Gini coefficient lies in the [0, l] ' t m erval), so that application of standard asymptotic results may lead to estimated intervals that extend beyond the theoretical bounds of a particular measure (e.g. a negative lower bound for Gini). An alternative method for computing probability intervals is to bootstrap. The bootstrap provides interval estimates drawn from the small sample distribution. These interval estimates have been shown to be superior to asymptotic intervals both the-3 oretically and in a variety of applications.. Bootstrap intervals are computationally inexpensive and easy to calculate, the same method applies to all the inequality measures used in the literature, and the bootstrap method automatically takes into account any bounds that, apply to a particular measure. Further, since bootstrap intervals computed using the percentile met.hod have a clear Bayesian interpretation, they provide a straightforward solution to the Behrens-Fisher problem of comparing means from two distributions (see section 3).

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The Measurement and Decomposition of Multi-Dimensional Inequality

Cited by 427 publications

References 14 publications

Constructing indices of multivariate polarization

Constructing indices of multivariate polarization

Statistical Inference via Bootstrapping for Measures of Inequality

Statistical Inference via Bootstrapping for Measures of Inequality

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