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Elementary multivariate rearrangements and stochastic dominance on a Fréchet class

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  • Decancq, Koen

Abstract

A Fréchet class collects all multivariate joint distribution functions that have the same marginals. Members of a Fréchet class only differ with respect to the interdependence between their marginals. In this paper, I study orders of interdependence on a Fréchet class using two multivariate generalizations of the bivariate rearrangement proposed by Epstein and Tanny (1980) [4] and Tchen (1980) [16]. I show how these multivariate rearrangements are underlying multivariate first order stochastic dominance in terms of the joint distribution function and the survival function. A combination of both rearrangements is shown to be equivalent to the concordance order proposed by Joe (1990) [9].

Suggested Citation

  • Decancq, Koen, 2012. "Elementary multivariate rearrangements and stochastic dominance on a Fréchet class," Journal of Economic Theory, Elsevier, vol. 147(4), pages 1450-1459.
  • Handle: RePEc:eee:jetheo:v:147:y:2012:i:4:p:1450-1459
    DOI: 10.1016/j.jet.2011.11.001
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    References listed on IDEAS

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    1. Joe, Harry, 1990. "Multivariate concordance," Journal of Multivariate Analysis, Elsevier, vol. 35(1), pages 12-30, October.
    2. Meyer, Margaret & Strulovici, Bruno, 2012. "Increasing interdependence of multivariate distributions," Journal of Economic Theory, Elsevier, vol. 147(4), pages 1460-1489.
    3. Nicolas Gravel & Patrick Moyes, 2006. "Ethically Robust Comparisons of Distributions of Two Individual Attributes," IDEP Working Papers 0605, Institut d'economie publique (IDEP), Marseille, France, revised Aug 2006.
    4. Müller, Alfred & Scarsini, Marco, 2000. "Some Remarks on the Supermodular Order," Journal of Multivariate Analysis, Elsevier, vol. 73(1), pages 107-119, April.
    5. D'Agostino, Marcello & Dardanoni, Valentino, 2009. "The measurement of rank mobility," Journal of Economic Theory, Elsevier, vol. 144(4), pages 1783-1803, July.
    6. Larry G. Epstein & Stephen M. Tanny, 1980. "Increasing Generalized Correlation: A Definition and Some Economic Consequences," Canadian Journal of Economics, Canadian Economics Association, vol. 13(1), pages 16-34, February.
    7. Marco Scarsini, 1988. "Dominance Conditions for Multivariate Utility Functions," Management Science, INFORMS, vol. 34(4), pages 454-460, April.
    8. A. B. Atkinson & F. Bourguignon, 1982. "The Comparison of Multi-Dimensioned Distributions of Economic Status," Review of Economic Studies, Oxford University Press, vol. 49(2), pages 183-201.
    9. David A. Hennessy & Harvey E. Lapan, 2003. "A Definition of 'More Systematic Risk' with Some Welfare Implications," Economica, London School of Economics and Political Science, vol. 70(279), pages 493-507, August.
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    Cited by:

    1. Gravel, Nicolas & Moyes, Patrick, 2012. "Ethically robust comparisons of bidimensional distributions with an ordinal attribute," Journal of Economic Theory, Elsevier, vol. 147(4), pages 1384-1426.
    2. Marling, Tina Gottschalk & Range, Troels Martin & Sudhölter, Peter & Østerdal, Lars Peter, 2018. "Decomposing bivariate dominance for social welfare comparisons," Mathematical Social Sciences, Elsevier, vol. 95(C), pages 1-8.
    3. Christoph Heinzel, 2014. "Term structure of discount rates under multivariate s-ordered consumption growth," Working Papers SMART - LERECO 14-01, INRAE UMR SMART-LERECO.
    4. Rolf Aaberge & Anthony B. Atkinson & Sebastian Königs, 2018. "From classes to copulas: wages, capital, and top incomes," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 16(2), pages 295-320, June.
    5. Benoît Tarroux, 2015. "Comparing two-dimensional distributions: a questionnaire-experimental approach," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 44(1), pages 87-108, January.
    6. Koen Decancq, 2020. "Measuring cumulative deprivation and affluence based on the diagonal dependence diagram," METRON, Springer;Sapienza Università di Roma, vol. 78(2), pages 103-117, August.
    7. Gajdos, Thibault & Weymark, John A., 2012. "Introduction to inequality and risk," Journal of Economic Theory, Elsevier, vol. 147(4), pages 1313-1330.
    8. Meyer, Margaret & Strulovici, Bruno, 2012. "Increasing interdependence of multivariate distributions," Journal of Economic Theory, Elsevier, vol. 147(4), pages 1460-1489.
    9. Aaberge, Rolf & Peluso, Eugenio & Sigstad, Henrik, 2019. "The dual approach for measuring multidimensional deprivation: Theory and empirical evidence," Journal of Public Economics, Elsevier, vol. 177(C), pages 1-1.
    10. Rolf Aaberge & Eugenio Peluso, 2011. "A Counting Approach for Measuring Multidimensional Deprivation," Working Papers 07/2011, University of Verona, Department of Economics.
    11. Kobus, Martyna & Kurek, Radosław, 2018. "Copula-based measurement of interdependence for discrete distributions," Journal of Mathematical Economics, Elsevier, vol. 79(C), pages 27-39.
    12. Christoffer Sonne-Schmidt & Finn Tarp & Lars Peter Østerdal, 2016. "Ordinal Bivariate Inequality: Concepts and Application to Child Deprivation in Mozambique," Review of Income and Wealth, International Association for Research in Income and Wealth, vol. 62(3), pages 559-573, September.
    13. Koen Decancq, 0. "Measuring cumulative deprivation and affluence based on the diagonal dependence diagram," METRON, Springer;Sapienza Università di Roma, vol. 0, pages 1-15.
    14. Frank A Cowell & Martyna Kobus & Radoslaw Kurek, 2017. "Welfare and Inequality Comparisons for Uni- and Multi-dimensional Distributions of Ordinal Data," STICERD - Public Economics Programme Discussion Papers 31, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    15. Christoffer Sonne-Schmidt & Finn Tarp & Lars Peter Østerdal, 2013. "Ordinal Multidimensional Inequality," WIDER Working Paper Series wp-2013-097, World Institute for Development Economic Research (UNU-WIDER).
    16. Rolf Aaberge & Eugenio Peluso & Henrik Sigstad, 2015. "The dual approach for measuring. Multidimesional deprivation and poverty," Discussion Papers 820, Statistics Norway, Research Department.
    17. Sonne-Schmidt, Christoffer & Tarp, Finn & Østerdal, Lars Peter, 2013. "Ordinal Multidimensional Inequality," WIDER Working Paper Series 097, World Institute for Development Economic Research (UNU-WIDER).
    18. Martyna Kobus & Radoslaw Kurek, 2017. "Copula-based measurement of interdependence for discrete distributions," Working Papers 431, ECINEQ, Society for the Study of Economic Inequality.
    19. Suman Seth and Maria Emma Santos, 2018. "Multidimensional Inequality and Human Development," OPHI Working Papers ophiwp114_2.pdf, Queen Elizabeth House, University of Oxford.
    20. Francesco Andreoli & Claudio Zoli, 2020. "From unidimensional to multidimensional inequality: a review," METRON, Springer;Sapienza Università di Roma, vol. 78(1), pages 5-42, April.

    More about this item

    Keywords

    Concordance order; Fréchet class; Multivariate rearrangements; Multivariate stochastic dominance; Orthant dependence order; Supermodular order;

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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