Measuring group disadvantage with inter-distributional inequality indices: A critical review and some amendments to existing indices

Yalonetzky, Gaston Article Measuring group disadvantage with inter-distributional inequality indices: A critical review and some amendments to existing indices Economics: The Open-Access, Open-Assessment E-Journal Provided in Cooperation with: Kiel Institute for the World Economy – Leibniz Center for Research on Global Economic Challenges Suggested Citation: Yalonetzky, Gaston (2012) : Measuring group disadvantage with interdistributional inequality indices: A critical review and some amendments to existing indices, Economics: The Open-Access, Open-Assessment E-Journal, ISSN 1864-6042, Kiel Institute for the World Economy (IfW), Kiel, Vol. 6, Iss. 2012-9, pp. 1-32, https://doi.org/10.5018/economics-ejournal.ja.2012-9 This Version is available at: https://hdl.handle.net/10419/56493 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by-nc/2.0/de/deed.en Measuring Group Disadvantage with Inter-distributional Inequality Indices: A Critical Review and Some Amendments to Existing Indices Gaston Yalonetzky University of Leeds Abstract A long literature on inter-distributional inequality (IDI) has developed statistical tools for measuring the extent of inequality between two groups (e.g. men versus women). The paper reviews some of the most prominent IDI indices proposed in the last four decades. The assessment focuses on how these indices react to inequalities that are disadvantageous to different groups, using two operationalizations of a concept of group-specific disadvantage focus (GDF). Relying on a complementary set of properties, the review also assesses whether these indices are informative about other interesting features related to IDI comparisons, chiefly distributional equality, but also absence of distributional overlap and presence of firstorder stochastic dominance. The author proposes amendments to several of these indices in order to render them in fulfillment of GDF properties and more informative on the mentioned distributional features. Special Issue The Measurement of Inequality and Well-Being: New Perspectives JEL D30, J71 Keywords Inter-distributional inequality Correspondence Gaston Yalonetzky, Maurice Keyworth Building, Leeds University Business School, Leeds LS2 9JT, United Kingdom, e-mail: [email protected]. Citation Gaston Yalonetzky (2012). Measuring Group Disadvantage with Inter-distributional Inequality Indices: A Critical Review and Some Amendments to Existing Indices. Economics: The Open-Access, Open-Assessment EJournal, Vol. 6, 2012-9. http://dx.doi.org/10.5018/economics-ejournal.ja.2012-9 © Author(s) 2012. Licensed under a Creative Commons License - Attribution-NonCommercial 2.0 Germany Vol. 6, 2012-9 | March 30, 2012 | http://dx.doi.org/10.5018/economics-ejournal.ja.2012-9 conomics: The Open-Access, Open-Assessment E-Journal 1 Introduction The concern for differences in the distribution of wellbeing characteristics among groups within societies has earned a long-standing interest in the Social Sciences and Political Philosophy. This concern has often emphasized the potential presence of socio-economic discrimination of different natures (e.g. Becker, 1971; Phelps, 1972; Arrow, 1973). In general, it has been associated with concepts of inequality of opportunities.1The normative view for between-groups differences, related to ethnicity or gender, states that they are intrinsically unfair (particularly when the groups are dened over characteristics beyond their members' control), and instrumentally detrimental to individuals and societies (e.g. Arneson, 1989; Cohen, 1989; Nussbaum and Glover, 1995; Roemer, 1998; Fleurbaey, 2001; Sen, 2001). From a quantitative perspective, one way of measuring the extent of differences in wellbeing between groups is to use indices that capture between-group inequalities, and that declare the total absence of between-group inequality when conditional distributions of wellbeing, or some functions of them, are identical across groups.2There is also an interest in quantifying between-group inequalities with a focus on capturing inequality if and when it is (more) detrimental to one specic group as opposed to other(s), i.e. a concept of relative economic disadvantage. Even though several authors have focused on inequalities detrimental to one group,3only recently formal denitions of the concept have been put forward. Interestingly, these recent denitions emphasize a concern for censoring inequal1For a good review of the literature on inequality of opportunity see Fleurbaey (2008). Also Roemer (1998). 2This condition is consistent with a literalist denition of inequality of opportunity by Roemer (1998, p. 15-6) as well as with Van De Gaer's rule (Ooghe et al., 2007). It is also consistent with Fleurbaey's concept of circumstance neutralization (Fleurbaey, 2008, p. 25). There are alternative ways of measuring between-group inequality. For instance, it could be measured as the residual inequality after within-group inequality has been suppressed (e.g. by replacing individual's wellbeing values with those of their group mean). Such approach has been followed, among others, by Roemer (2006); Elbers et al. (2008); Ferreira and Gignoux (2011); Lanjouw and Rao (2011). 3This literature is abundant. Some important examples are Gastwirth (1975), Butler and McDonald (1987), Dagum (1987), Jenkins (1994), van Krem (2009), Gradin et al. (2010) and del Rio et al. (2011). www.economics-ejournal.org 1 conomics: The Open-Access, Open-Assessment E-Journal ities when they are not detrimental to the group of concern. The most recent and neat denition by del Rio et al. (2011), based on the work of Jenkins (1994), applies to comparisons of actual distributions against counterfactuals. This approach effectively deals with distributions of the same population size. This paper provides a review of inter-distributional inequality (IDI) indices. Considering the renewed interest in measuring between-group inequalities with a focus on those which are detrimental to one specic group, the review explores how we can measure inequalities with metrics that are exclusively sensitive to only one specic group's disadvantage. Since there are several indices of interdistributional inequality (IDI) already available, I propose some ways of measuring this focused inequality by suggesting some amendments to existing indices, which do not measure IDI with a focus on specic disadvantages in their current forms. Conducting the review with such a concern for exclusive sensitivity to inequality detrimental to one group, is also helpful for understanding, and clarifying, how existing IDI indices deal with, and react to, inequalities that are detrimental to different groups. With these two purposes in mind, the paper begins with a discussion of the concept of group-specic disadvantage focus (GDF). Since there are different plausible ways of operationalizing this concept in an axiom, I propose two options for a property of a(n index's) sensitivity to inequality that is detrimental exclusively to one specic group. Both properties are applicable to IDI indices that deal with populations of different size. The rst property is called quantile group-specic disadvantage focus (QGDF). The second property is called overlap group-specic disadvantage focus (OGDF). An advantage of the rst option is that it can be related to indices that measure inequality on quantile space, or on probability space. This review of existing indices also evaluates whether they are informative, or not, regarding other interesting features related to IDI comparisons. For instance, do they unambiguously pinpoint situations in which two distributions are identical? Remarkably most of them do not. Moreover, the two proposed operationalizations of GDF, i.e. QGDF and OGDF, are related, respectively, with rst-order stochastic dominance and the degree of distributional overlap. Hence I also assess whether the IDI indices are informative as to the absence of distributional overlap and/or presence of rst-order stochastic dominance. I propose www.economics-ejournal.org 2 conomics: The Open-Access, Open-Assessment E-Journal some further amendments that improve the indices' informative content on these features. There are several indices of inter-distributional differences. Therefore in this paper I focus on IDI indices that are characterized by: i) being useful specically for two-group comparisons, ii) being more informative than just comparing two means, and iii) being useful when the two distributions have different sample sizes.4I rst review the PROB index by Gastwirth (1975), followed by the closely related family of indices of relative distributions, discussed by Handcock (1999) and by Le Breton et al. (2008). These indices, which map from probability functions, do not fulll QGDF because in some cases they compensate inequalities detrimental to one group with inequalities detrimental to the other group, while in other cases they just add up the two forms of inequalities together. In relation to that, most of these indices do not distinguish situations of equal distributions from other situations wherein there is inequality. Finally, while several of these indices are helpful to pinpoint situations of lack of distributional overlap, they are not informative as to the presence of rst-order stochastic dominance. I propose some simple amendments to these indices that render them more informative about the aforementioned features; chiey, the extent of group-specic disadvantage. I then review the family of quantile-based indices of Ebert (1984) and Vinod (1985). As in the previous case, these indices do not fulll QGDF, either because they compensate group-specic detrimental inequalities or because they add them up indiscriminately. As for other features, while Ebert's index does differentiate, unambiguously, between distributional equality and other situations, Vinod's does not. Neither index is helpful to detect absolute lack of distributional overlap. I propose simple amendments to these indices that render them both in fulllment of QGDF, more informative in terms of presence of rst-order stochastic dominance, and in compliance with other properties, like scale invariance, which are desirable for certain IDI comparisons. 4When sample sizes are identical the literature on counterfactual comparisons, e.g. del Rio et al. (2011), provides the relevant indices. However, even without the explicit purpose, mobility indices may also be amendable to render them suitable for the analysis of between-group inequalities with GDF and identical populations. Good examples of such indices are provided by Cowell's measures of distributional change (Cowell, 1985), by Fields (1996); Fields and Ok (1999) and by Schluter and van de Gaer (2011). www.economics-ejournal.org 3 conomics: The Open-Access, Open-Assessment E-Journal Then I turn to an assessment of the family of indices proposed by Dagum (1980, 1987). These indices compare each value of the wellbeing variable in one group against all the values present in the other group. The Dagum family does not fulll QGDF and does not distinguish a situation of distributional equality from other cases of inequality. However the Dagum family hosts the best examples of indices satisfying OGDF. Accordingly they are useful in pinpointing situations of absence of distributional overlap. The next family under review comprises the indices based on incomplete moments (Butler and McDonald, 1989). These include the indices by Butler and McDonald (1987) and those by Deutsch and Silber (1997). The review shows that these indices do not fulll QGDF. However, with amendments, some of them can fulll QGDF, while some others can fulll OGDF. The review also shows the close relationship between these indices and those of the Dagum family. Indices based on incomplete moments are not informative about distributional equality or rst-order stochastic dominance. However, they are also useful for the identication of absence of distributional overlap. Finally, I complete the review with an appraisal of the family of ethical distance indices proposed by Shorrocks (1982), and axiomatically characterized by Chakravarty and Dutta (1987). Ethical distance indices are different from the previous ones in that they compare equallydistributed-equivalent (EDE) standards from the distributions.5This requires a rst aggregation step in which each distribution is mapped into its respective EDE standard. Then two such standards are compared. Despite this difference, I include these indices in the review because they have been proposed as alternatives to, and contrasted with, some IDI indices (see Shorrocks, 1982). I explain why, notwithstanding their merit and appeal, this family of indices does not fulll notions of GDF. The indices are also of little help for pinpointing situations of distributional equality, rst-order stochastic dominance and/or absolute absence of distributional overlap. The next section introduces the basic notation and a minimum set of properties that IDI indices are expected to fulll. The main subsection denes the two properties that operationalize the concept of group-specic disadvantage focus. Then the review and proposal of new amendments is done in subsequent sections: 5EDE standards were introduced by Atkinson (1970). www.economics-ejournal.org 4 conomics: The Open-Access, Open-Assessment E-Journal one for the PROB index and indices based on relative distributions; followed by a section on the quantile indices; then followed by a section on the Dagum family, a section on incomplete moment indices, and a section on ethical distance indices. The paper ends with some concluding remarks. 2 Notation and Basic Properties Consider two population groups, one with distribution Xand size M, and the other one with distribution Yand size N. Hence X(M):= (x1;x2;:::;xM)and Y(N):= (y1;y2;:::;yN). Group sizes can be different. The density function of Xis fX(z) and its cumulative distribution function (cdf) is FX(z), where zis a wellbeing continuous variable. As usual, Z∞ ∞fX(z)dz =1 and FX(z) = Zz ∞fX(s)ds. The inverse of the cdf yields the quantiles of X. These quantiles are dened as: x(p) F1 X(p), where p2[0;1]. Effectively, p=FX(x(p)). An IDI index, D(X;Y), maps from RMRNto the real line. Now the rst two properties that are reasonable for IDI indices are population invariance (or principle of population) and scale invariance, both traditional axioms from the wellbeing measurement literature. In the case of two group distributions, an IDI index is said to fulll population invariance if and only if its value is not affected by an identical replication of members within each group, although the number of replications can vary between groups:D(X(M);Y(N)) = D(X(λMM);Y(λNN)), where λMand λNare two different scalars. An IDI index is said to fulll scale invariance if and only if its value is not affected by multiplying all the values of both distributions by the same scalar; i.e. D(X(M);Y(N)) = D(λx1;:::;λxM;λy1;:::;λyN), where λis a scalar. Fulllment of these two properties ensures that the IDI comparison is not affected either by changes in relative population sizes per se, or by the unit of measurement used to quantify the wellbeing attribute (e.g. income expressed in different currencies). An alternative, or complement, to scale invariance is translation invariance (e.g. see Ebert, 1984, axiom 2; and Magdalou and Nock, 2011). An IDI index fullls translation invariance if and only if its value is not www.economics-ejournal.org 5 conomics: The Open-Access, Open-Assessment E-Journal affected by adding the same scalar to all the values of both distributions, i.e. if D(X(M);Y(N)) = D(x1λ;:::;xMλ;y1λ;:::;yNλ): The next desirable property is related to the ability of an IDI index to identify situations of distributional equality (DE). DE holds if and only if fX(z) = fY(z)8z. DE can also be expressed in terms of cumulative distribution functions (i.e. FX(z) = F Y(z)8z), or in terms of quantiles (i.e. x(p) = y(p)8p2[0;1]). Whenever, in the literature, the index is required to be sensitive to the presence of DE, it is designed to take its minimum value under DE, which is usually zero. However this property of sensitivity to DE can take a weak form and a strong form. The weak form of the property is the following: Axiom 1 Weak Sensitivity to Distributional Equality (WSDE): An IDI index is weakly sensitive to distributional equality if: fX(z) = fY(z)8z!D(X;Y) = 0. Axiom 1 is basically property (2a) in Shorrocks (1982). WSDE requires the index to take its minimum value (zero) whenever there is distributional equality. However, in principle, an index satisfying WSDE could take that same value in alternative situations of distributional inequality. Hence, for an IDI index to be most informative regarding the presence of DE, it should take its minimum value only when DE holds. That is, it should fulll the following property: Axiom 2 Strong Sensitivity to Distributional Equality (SSDE): An IDI index is strongly sensitive to distributional equality if: fX(z) = fY(z)8z$D(X;Y) = 0. SSDE is Ebert's reexivity property (but expressed in terms of densities; Ebert, 1984, p. 268). A focus on Group-specic Disadvantage It is much easier to dene a situation of DE than to characterize all the different possible forms of IDI, even though the former is rarely observed in practice. Most of the literature on IDI comparisons based on indices for distributions with different population size, has taken one of two conceptual approaches to measure IDI. One approach is to measure IDI as a cumulative departure from DE. In this approach, between-group distributional differences are aggregated without www.economics-ejournal.org 6 conomics: The Open-Access, Open-Assessment E-Journal distinguishing whether these differences are favourable, or not, to any specic group. That's the route followed by Ebert (1984) and Chakravarty and Dutta (1987) . A property of symmetry, whereby the indices are unaffected by switching the two distributions around, is usually advocated in this rst approach; i.e. D(X;Y) = D(Y;X). A second approach acknowledges, more explicitly, that some distributional differences can be said to favour one specic group over another one. But then the measures of these differences, quantifying relative advantage for each group respectively, are pitted against each other, in order to derive an index of net advantage. This is the approach followed by Butler and McDonald (1987) , Vinod (1985) and Dagum (1987) , among others. Recently, on the other hand, the literature on discrimination measurement based on counterfactual comparisons is advocating a third approach: indices that are sensitive only to distributional differences that are favourable (or detrimental) to one group in particular (e.g. del Rio et al., 2011). This approach is not new. For instance, the contribution of Dagum (1980) was already in that spirit. However, there has not been an exhaustive discussion of how to operationalize the notions of group-specic disadvantage focus (GDF), i.e. an exclusive sensitivity to inequalities that are detrimental to one specic group, as advocated by the literature on counterfactual comparisons. Without claiming, or aiming, to explore all the possible options, in this section I propose two intuitive and meaningful ways of operationalizing the concept of GDF. These are used in the rest of the review to assess, and better understand, the behaviour of the IDI indices. In order to introduce the rst operationalization, QGDF, it is worth starting by noting that two distributions may be different in many ways. For instance, they may have different means. Or even if they have equal means, they may differ in their average spread, skewness or kurtosis. More importantly, from a wellbeing perspective, these inter-distributional differences may render one distribution more desirable than the other one as a "lottery". The stochastic dominance literature discusses this type of partial-ordering comparisons. But even when stochastic dominance relationships do not hold over the whole admissible range of a wellbeing variable, one may be able to make statements about whether certain parts of a distribution are more advantageous for one group vis-a-vis another one. For instance, consider income distributions A and B. Both are symmetric and www.economics-ejournal.org 7 conomics: The Open-Access, Open-Assessment E-Journal latter. An additional proposal can be made by combining PROB1 Y(YX)and PROB1 Y(XY)with AAD : RY(YX) = PROB1 Y(YX) 2AAD (8) RY(XY) = PROB1 Y(XY) 2AAD (9) RY(YX)provides a measure of the proportion of the inter-distributional inequality that is detrimental to Y;when the distribution of Yis taken as reference. An example of its usefulness is provided by the two cases in Figure 1. PROB1 Y(XY)yields the same value for both cases. By contrast, RY(XY) = 1 for the case of the left panel, whereas RY(XY)<1 for the case of the right panel. Figure 1: Left panel: the two CDFs overlap in part of their common support. Right panel: the two CDFs cross once www.economics-ejournal.org 14 conomics: The Open-Access, Open-Assessment E-Journal Handcock (1999) make a few further suggestions for indices based on cumulative relative distributions, and inspired by measures of goodness of t. One such index is the following, based on the stastistic used in the Cramer-von Mises test: CM =Z1 0GX=Y(F Y)F Y2dFY(10) Again, note that 3CM =PROB1 Y(YX) + PROB1 Y(XY). Hence CM behaves similarly to AAD, i.e it does not fulll QGDF because it adds the two types of inequalities; but it fullls SSDE. Likewise, it is not informative about rst-order dominance, whereas it is informative about lack of overlap since: CM =1 3$ FXzY min=1:Handcock (1999) also suggest using several divergence measures from the statistic literature, but mapping from ratios of the density functions (see their Table 5.1, p. 65). For instance, two measures from that list, which they focus on, are: Dχ=Z1 0 (fX fY1)dFY(11) DKL =Z1 0 log(fX fY )fX fY dF Y(12) Dχis inspired by Pearson's chi-square statistic whereas DKL is based on the Kullback-Leibler divergence measure. Unlike other measures based on relative distributions, it is not easy to render divergence measures like (11) and (12) (and those in Table 5.1 in Handcock, 1999) in fulllment of any operationalization of GDF due to their mapping from density functions. On the other hand, all these measures satisfy SSDE, i.e. they are good at pinpointing situations of distributional equality. In fact, they have long been used to test the equality of two distributions. Besides distributional equality, these measures are not informative of other distributional features of interest (e.g. stochastic dominance). Moreover, when applied to continuous variables, their computation requires techniques based on Kernel densities (Handcock, 1999). Hence, for the purpose of IDI comparisons based on indices mapping from probabilities, those that rely on cumulative probabilities should be preferred to those depending on densities. www.economics-ejournal.org 15 conomics: The Open-Access, Open-Assessment E-Journal 4 The Indices by Ebert and Vinod: Review and Amendments In a seminal contribution Ebert (1984) axiomatically characterized a family of indices based on Minkowski distances that is useful for IDI measurement. These indices are direct functions of the percentile gaps. The family for two groups with different population sizes is: dr(X;Y)Z1 0jy(p)x(p)jrdp1 r 8r1 (13) This proposal is similar to that of Vinod (1985) in that both are direct functions of the percentile gaps. Vinod's measure of "overall economic advantage" is: V(X;Y)Z1 0 [y(p)x(p)]d p =µYµX;(14) where µYis the mean of distribution Y.11 It is easy to check that both drand Vdo not fulll QGDF. As in the case with AAD,dris sensitive to both types of inequalities, which are added up by the index. By contrast, Vcompensates them. For that reason Vfullls WSDE but not SSDE, i.e. it is not useful to pinpoint distributional equality; whereas, like AAD,drsatises SSDE (property 2a in Ebert, 1984). Here it is worth noting that the lack of fulllment of QGDF, by dr, is a logical consequence of the index's fulllment of a symmetry property which Ebert (1984) adapted from Shorrocks (1982): An IDI index satisfying symmetry should not take a different value when the distributions of Xand Yare switched around. Clearly, the property of symmetry rules out any form of GDF. Neither drnor Vare informative regarding situations of rst-order stochastic dominance or absence of overlap. However simple amendments, which relate to both drand V, fulll QGDF and, combined, provide more information about distributional equality and rst-order dominance. The two amended indices are: 11 Vinod also considered partial measures of economic advantage, e.g. computations of (14) in a restricted quantile range. However, unlike Ebert, Vinod did not characterize his measures axiomatically. www.economics-ejournal.org 16 conomics: The Open-Access, Open-Assessment E-Journal dr YXZ1 0jy(p)x(p)jr +dp 8r1 (15) dr XYZ1 0jx(p)y(p)jr +dp 8r1 (16) Clearly, both (15) and (16) fulll QGDF, at the expense of symmetry. Together these indices also pinpoint DE because: dr YX=dr XY=0$fX=fY. They also detect rst-order dominance since:dr YX>0^dr XY=0$XFD Y:However neither the amendments nor the original indices take specic values if and only if there is absence of overlap; except, in the odd case for dr(X;Y), when either y(p) or x(p)are equal to zero for all p. Finally the amendments are related to drand Vaccording to the following expressions: [dr(X;Y)]r=dr YX+dr XY8r1 (17) V(X;Y) = d1 YXd1 XY(18) Unlike the indices in the previous section, drand Vare not bounded from above and do not fulll scale invariance. The latter, in the case of dr, is directly due to Ebert's requirement that his indices fulll a property of linear homogeneity, whereby multiplying all values of Xand Yby a common scalar should translate into a multiplication of drby the same scalar (Ebert, 1984, Axiom 1, p. 269). Clearly, V, (15) and (16), also satisfy linear homogeneity. Such property may not always be desirable. For instance, if one does not want an IDI comparison over income to be affected by the choice of currency. However scale invariance can be met, in conjunction with a normalization property that caps the indices from above, by dividing the indices by their maxima: ddr(X;Y)dr(X;Y) Z1 0 y(p)rdp1 r +Z1 0 x(p)rdp1 r (19) VV (X;Y)V(X;Y) µY+µX (20) www.economics-ejournal.org 17 conomics: The Open-Access, Open-Assessment E-Journal The amended indices, (15) and (16), can be normalized the same way as in (19). Of course, in all cases like (19) and (20), fulllment of linear homogeneity is relinquished. All indices in this section also fulll population invariance and translation invariance (Ebert, 1984, Axiom 2, p. 269). 5 The IDI Indices of Dagum Dagum's rich family of indices deserves special attention for two reasons. Firstly, in the IDI literature working with continuous variables and different distributional sizes, it is the only family whose indices map explicitly from differences in values from the two distributions, e.g. yx. Secondly, some of the indices provide a clear example of fulllment of some operationalizations of GDF, but not others. In this section I start reviewing Dagum's later contributions rst, because, as shown in the section, his earlier contributions are actually better suited to capture notions of GDF. Hence, in a sense, these can work as amendments of the later contributions, if the concern is to measure IDI with some operationalization of GDF. Dagum (1987) proposed a measure of relative economic afuence (REA) which, in this paper's notation, is dened as: DX=Y=1dY dX, where: dX=Z∞ 0 dFY(y)Zy 0 (yx)dFX(x);(21) dY=Z∞ 0 dFX(x)Zx 0 (xy)dFY(y):(22) DX=Yfullls population invariance, translation invariance and scale invariance, but it does not fulll either OGDF or QGDF. With respect to OGDF, it is clear that DX=Ydepends on yxwhen y>x and on xywhen x>y. As for QDF, rst, note that dXdY=µYµX.12 Hence: DX=Y=µYµX dX=V(X;Y) dX. This means that Dcompensates quantile gaps that are detrimental to different groups in the same way that Vdoes. So neither can fulll QGDF. Like V,Dfullls WSDE but not SSDE, i.e. it is not useful to pinpoint distributional equivalence, since the necessary and sufcient requirement for DX=Y=0 is: µX=µY. 12 This result stems from equations (4) and (5) of Dagum (1987, p. 6). www.economics-ejournal.org 18 conomics: The Open-Access, Open-Assessment E-Journal DX=Ydoes not identify situations of rst-order dominance because, even though XFD Yimplies dYdX, the reverse is not true. By contrast, DX=Yis useful for pinpointing absence of overlap. When the richest person in Yis poorer than the poorest person in Xthen dX=0 (and dY=µXµY) and the reverse is also true. When the richest person in Xis poorer than the poorest person in Y then dY=0 (and dX=µYµX) and the reverse is also true. dXcompares every member of Yagainst all the people in Xwho have less income and quanties the respective gaps. That is why the measures are well suited to detect absence of overlaps without resorting to quantiles or probabilities: if everybody in Xis richer than everybody in Ythen dX=0. The reverse is true because, unless the two distributions are degenerate and equal to each other, dX=0 requires that fX(y) = 0 for every value of yon the support of Y(and then FX(y) = 0 over the same support). The most straightforward alternatives to DX=Ythat may fulll some forms of GDF, are using its basic constituent statistics, i.e. dXand dY. In fact, in an earlier contribution, Dagum (1980) proposed using dX, or dY, as the basic statistics for measures of economic distance normalized by their respective minima and maxima.13 Now, clearly, dXand dYfulll OGDF; and, as other measures fullling the notions of GDF considered in this review, they are not symmetric. Like DX=Y, dXand dYdo not identify situations of rst-order stochastic dominance, but they are good for detecting absence of distributional overlap due to the aforementioned reasons. 13 More generally, Dagum also suggested considering the following family of statistics based on generalized means, even though he focused on d1 X: dr X=Z∞ 0 dF Y(y)Zy 0(yx)rdFX(x)1 r ;r6=0 (23) d0 X=eR∞ 0dFY(y)Ry 0ln(yx)dFX(x):(24) www.economics-ejournal.org 19 conomics: The Open-Access, Open-Assessment E-Journal However dXdoes not fulll QGDF. For instance, following Shorrocks (1982), dXcan be decomposed in the following way: dX=V(X;Y) 2+1 2Z1 0Z1 0jy(py)x(px)jdpxd py;(25) where pxand pyare percentiles of Xand Y, respectively. Hence dXcompensates and adds up quantile gaps that are detrimental to diffferent groups. Likewise, the family of generalized means, i.e. (23) and (24), fullls OGDF, but not QGDF. The reason for the latter is that the difference yx, in the respective formulas, can be expressed as: yx=y(py)y(px)+y(px)x(px), using the notation introduced in (25). Hence, even though y>x, in some cases y(px)>x(px), whereas in others y(px)<x(px). Therefore quantiles gaps that are detrimental to different groups are compensated, contrary to the requirements of QGDF. Inability to fulll QGDF should not be considered a serious drawback for dX, even if one is interested in IDI indices sensitive to some notion of GDF, because dXdoes capture alternative meaningful concepts of GDF, e.g. OGDF. By contrast, a problematic feature of this index is its inability to fulll even WSDE. As shown by Shorrocks (1982), when there is distributional equality dX=dY=µXG(X) = µYG(Y), where G(X)is the Gini coefcient of X. However the reverse is not true because two different distributions can have the same mean and Gini coefcient. For instance if Yis obtained from Xby performing two transfers of the same amount, but one regressive and one progressive, involving two pairs of individuals in different parts of the distribution, then both distributions, despite being unequal, have the same mean and the same value for the Gini coefcient. 14 Unlike DX=Y,dXdoes not fulll scale invariance. Nor it is normalized. However, the following amendment of (21) fullls scale invariance and is normalized so that it is equal to 1 if and only if FXzY min=1, and it is equal to 0 if and only if FXzY max=0: DNX=dX dX+dY :(26) 14 For instance if X= (1;2;3;4)and Y= (0:5;2:5;3:5;3:5). www.economics-ejournal.org 20 conomics: The Open-Access, Open-Assessment E-Journal A similar amendment is applicable to (24). Interestingly, when µY=µX: DNX=DNY=0:5 6 IDI Indices Based on Incomplete Moments Some authors have proposed measures based on incomplete moments for IDI comparisons. Incomplete moments take the following form: φ(x;h) = Zx 0 yhdF Y(y) E(yh);(27) where EyhZ∞ 0 yhdF Y(y). Of the handful of indices proposed, I review the three that use more information from the cumulative distributions of the two compared groups. As this section shows, there is a close connection between some of these indices and the PROB index, and also with the Dagum family. These links make it easier to ascertain which properties are fullled by indices based on incomplete moments. The rst index is P(1;1), one from a group of Pietra indices proposed by Butler and McDonald (1987) : P(1;1) = ZF Y(µX) 0 y(p)dp µYZFX(µY) 0 x(p)dp µX (28) P(1;1)measures the difference between the proportion of total income in Yheld by people who have income not higher than the average income in X minus the proportion of total income in Xheld by people who have income not higher than the average income in Y. Even though the index is not symmetric, it can be shown that P(1;1)does not fulll QGDF. In order to prove this, imagine that distributions Yand Xare both symmetric with equal mean, µ, hence: F Y(µX) = FX(µY). In that situation: P(1;1) = Z0:5 0 [y(p)x(p)]dp µ:Yet the gaps www.economics-ejournal.org 21 conomics: The Open-Access, Open-Assessment E-Journal y(p)x(p)can have different signs in the integration interval (e.g. imagine the two distributions differ in their kurtosis). Hence P(1;1)may compensate quantile gaps that are detrimental to different groups. With a bit more manipulation it is also possible to prove that P(1;1)does not fulll OGDF either in the case of equal means (µ). The key is to reexpress (28), with equal means, as: P(1;1) = 1 µ[Z∞ 0Zµ 0 (yx)dFY(y)dFX(x)+Z∞ 0Zµ 0 (yx)dFX(x)dFY(y)+ Z∞ 0 xF Y(µ)dFX(x)Z∞ 0 yFX(µ)dFY(y)] (29) Expression (29) is, then, sensitive to gaps yxof different signs. Hence OGDF is not satised. Now, because P(1;1)may compensate quantile gaps, it does not fulll SSDE, although it does satisfy WSDE. Likewise it is not difcult to nd examples showing that the measure is not helpful in identifying rst-order stochastic dominance either. In the absence of overlap P(1;1) = 1 if the poorest person in Xis richer than the richest person in Y, and P(1;1) = 1 if the poorest person in Yis richer than the richest person in X. However the reverse relationships are not true. For instance, it sufces for P(1;1) = 1 that the richest person in Yhas less than the mean income of Xand the poorest person in Xhas more than the mean income of Y. Amendments to P(1;1)that may render it in fulllment of notions of GDF, or more informative about the distributional features under discussion, do not seem to be straightforward. The second index has been proposed by Deutsch and Silber (1997). For continuous variables, it is: I0 G2=Z∞ 0 dFX(x)Zx 0 dF Y(y)Z∞ 0 dFY(y)Zy 0 dFX(x)(30) Interestingly, (30) is the difference between two of the d0measures of Dagum (1980). From the denition of PROB, it is easy to show that PROBY=1 www.economics-ejournal.org 22 conomics: The Open-Access, Open-Assessment E-Journal Z∞ 0 dFX(x)Zx 0 dF Y(y)and PROBX=1Z∞ 0 dFY(y)Zy 0 dFX(x).15 Hence I0 G2= PROBYPROBX. This means that I0 G2inherits the inability to fulll QGDF from the PROB measures, as is clear in the following expression: I0 G2=2[Z∞ 0 [F Y(z)FX(z)]+fY(z)dzZ∞ 0 [FX(z)F Y(z)]+fY(z)dz](31) Likewise I0 G2fullls WSDE but not SSDE. It also fails to pinpoint situations of rst-order stochastic dominance. By contrast, it is helpful for the detection of absence of overlap, since: PROBY=1$PROBX=0, in which case I0 G2=1 if and only if the poorest person in Xis richer than the richest person in Y. Similarly I0 G2=1 if and only if the poorest person in Yis richer than the richest person in X. Amendments to I0 G2in order to make it capture notions of GDF (e.g. QGDF) may lead to proposals similar to those in the above section discussing the PROB measures. Finally, the third index of incomplete moments reviewed has also been proposed by Deutsch and Silber (1997). For continuous variables, it is: I0 G1=1 µXµY [Z∞ 0 dFX(x)xZx 0 ydF Y(y)Z∞ 0 dFY(y)yZy 0 xdFX(x)] (32) With some manipulation, one can show that (32) is also equal to: I0 G1=1 µXµY [Z∞ 0 x2F Y(x)dFX(x)Z∞ 0 y2FX(y)dFY(y) Z∞ 0 dFX(x)xZx 0 (xy)dFY(y) +Z∞ 0 dFY(y)yZy 0 (yx)dFX(x)] (33) Now, with (33), a resemblance to Dagum's DX=Yis apparent. Both are sensitive to gaps yxwith different signs, and compensate for them. Hence I0 G1is 15 As shown also by Dagum (1980). www.economics-ejournal.org 23 conomics: The Open-Access, Open-Assessment E-Journal Ebert, U. (1984). Measures of distance between income distributions. Journal of Economic Theory, 32: 266–274. URL http://ideas.repec.org/a/eee/jetheo/ v32y1984i2p266-274.html. Elbers, C., Lanjouw, P., Mistiaen, J., and Ozler, B. (2008). Reinterpreting between-group inequality. Journal of Economic Inequality, 6: 231–245. 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