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Measuring cumulative deprivation and affluence based on the diagonal dependence diagram

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Abstract

In this paper, I introduce the diagonal dependence diagram to chart dependence around the main diagonal of a multivariate distribution. This diagonal dependence diagram is a useful tool to quantify phenomena such as cumulative deprivation and affluence. A society is said to exhibit more cumulative deprivation when more persons occupy bottom positions in all dimensions of well-being. Analogously, there is more cumulative affluence in a society when more persons occupy top positions in all dimensions. The diagonal dependence diagram consists of two curves which are obtained by taking the diagonal section of the underlying copula and survival functions, respectively. I show the elementary multivariate rearrangements underlying dominance in terms of both curves. The area under each curve leads to a natural index of diagonal dependence. Interestingly, the average of both indices equals a multivariate generalization of Spearman’s footrule and is closely related to the cograduation index proposed by Gini.

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Notes

  1. See [26, 27] for (bibliographic) overviews of Gini’s contribution to the field of inequality measurement.

  2. The statistical approach may be contrasted with the normative approach to the measurement of inequality, developed in the wake of [5, 14]. This normative approach focuses on the social welfare cost of inequality, rather than on inequality itself.

  3. For a non-exhaustive overview of the literature on multidimensional inequality measurement, see [8, 12, 31, 32, 56, 59]. Andreoli and Zoli [4] provide a recent review and study how results about unidimensional stochastic dominance can be extended to the multidimensional setting.

  4. The notion of cumulative deprivation generalizes the intersection approach that is used in the analysis of multidimensional poverty, see [2, 3, 6].

  5. The main diagonal connects the position vector that consists of the bottom position in all dimensions with the position vector that consists of the top position in all dimensions.

  6. See [22] for a historical overview of the use of Spearman’s footrule and its link to Gini’s cograduation index.

  7. See [38] and [55] for a thorough discussion of the copula function and its properties.

  8. For \(m\ge 3\), the function \(C_{-}\) is a lower bound for the copula function, but it is not a distribution function and, hence, cannot be a copula function [38, p. 47].

  9. In mathematical statistics, the downward diagonal dependence curve is known as the diagonal section of a copula function, see [38, p. 12]. Fernández-Sánchez and Úbeda-Flores [20] give a recent review of the statistical literature that deals with the existence and properties of the copula for a given diagonal section.

  10. To be precise, the upward diagonal dependence curve is the diagonal section of the survival copula \(\widehat{C}_{X}\left( p,\ldots ,p\right) =\overline{C}_{X}\left( 1-p,\ldots ,1-p\right) .\) On the definition of the survival copula \(\widehat{C}\) and its relation to the survival function \(\overline{C}\), see [38, pp. 32–33].

  11. Both curves coincide for random vectors which exhibit radial symmetry about \((1/2,\ldots ,1/2)\), so that \(C_{X}\left( p,\ldots ,p\right) =\overline{C}_{X}\left( 1-p,\ldots ,1-p\right) \) for all \(p=\left( p_{1},\ldots ,p_{m}\right) \) in \(\mathbb {I}\) [38, p. 36].

  12. Echoing the similarity between the downward diagonal dependence ordering and univariate first order stochastic dominance, the upward diagonal dependence ordering is formally similar to the univariate first order decumulative stochastic dominance discussed by Bazen and Moyes [10].

  13. When \(m=2,\) we have that \(\overline{C}_{X}\left( p,p\right) =1-2p+C_{X}\left( p,p\right) \).

  14. In the following, I adopt the convention that 0 is even.

  15. Müller [34] refers to these rearrangements as \(\varDelta \)-monotone transfers and \(\varDelta \)-antitone transfers, respectively.

  16. See [18, Lemma 3.1] and [57, Lemma 4.1] for a formal proof.

  17. This can be checked by observing that and .

  18. See [22, 41, 57] for more extensive discussions of Spearman’s footrule.

  19. Pérez and Prieto [40] advocate the use of an analogous decomposition of Spearman’s \(\rho \) in a “downward” measure \(\rho ^{-}\) and an “upward” measure \(\rho ^{+}\), which have been discussed by Joe [29], Nelsen [36], and Schmid and Schmidt [47].

  20. More recently, Dolati [18] and Taylor [52] discuss this symmetry as a desirable property of a multivariate measure of concordance.

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Authors and Affiliations

  1. Centre for Social Policy, University of Antwerp, Antwerp, Belgium

    Koen Decancq

  2. Department of Economics, KULeuven, Leuven, Belgium

    Koen Decancq

  3. Centre for Philosophy of Natural and Social Science, LSE, London, UK

    Koen Decancq

  4. CORE, Université catholique to Louvain, Louvain-la-Neuve, Belgium

    Koen Decancq

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I thank Begoña Cabeza, Shaun Da Costa, Vanesa Jordá, Ana Perez Espartero, Mercedes Prieto Alaiz, Giovanna Scarchilli, and Mateo Séré for useful comments. Financial support of the Research Foundation-Flanders is gratefully acknowledged.

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Decancq, K. Measuring cumulative deprivation and affluence based on the diagonal dependence diagram. METRON 78, 103–117 (2020). https://doi.org/10.1007/s40300-020-00173-7

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