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Increasing concave orderings of linear combinations of order statistics with applications to social welfare

  • Antonia Castaño-Martínez
  • Gema Pigueiras
  • Georgios Psarrakos
  • Miguel A. SordoEmail author


We provide in this paper sufficient conditions for comparing, in terms of the increasing concave order, some income random variables based on linear combinations of order statistics that are relevant in the framework of social welfare. The random variables under study are weighted average incomes of the poorest and, for some particular weights, their expectations are welfare measures whose integral representations are weighted areas underneath Bonferroni curves.


Increasing concave order Generalized Lorenz order Welfare measurement Order statistics 

JEL Classification




We thank the associated editor and two anonymous reviewers for constructive and helpful comments. M.A.S and G.P acknowledge support received from the Ministerio de Economía y Competitividad under Grant MTM2017-89577-P. A.C.M. acknowledges support received from the Ministerio de Economía y Competitividad under Grant MTM2016-74983-C2-2-R.

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


  1. Aaberge R (2000) Characterizations of Lorenz curves and income distributions. Soc Choice Welf 17:639–653CrossRefGoogle Scholar
  2. Arnold BC, Sarabia JM (2018) Majorization and the Lorenz order with applications in applied mathematics and economics. Springer, BerlinCrossRefGoogle Scholar
  3. Arnold BC, Balakrishnan N, Nagaraja HN (2008) A first course in order statistics. Classics in applied mathematics, vol 54. Society for Industrial and Applied Mathematics (SIAM), PhiladelphiaCrossRefGoogle Scholar
  4. Avérous J, Genest C, Kochar SC (2005) On the dependence structure of order statistics. J Multivariate Anal 94:159–171MathSciNetCrossRefGoogle Scholar
  5. Balakrishnan N, Belzunce F, Sordo MA, Suárez-Lloréns A (2012) Increasing directionally convex orderings of random vectors having the same copula and their use in comparing ordered data. J Multivariate Anal 105:45–54MathSciNetCrossRefGoogle Scholar
  6. Barlow RE, Proschan F (1975) Statistical theory of reliability and life testing. Holt, Rinehart and Winston, New YorkzbMATHGoogle Scholar
  7. Belzunce F, Martínez-Riquelme C, Mulero J (2016) An introduction to stochastic orders. Elsevier, AmsterdamzbMATHGoogle Scholar
  8. Bonferroni CE (1930) Elementi di statistica generale. Libreria Seber, FirenzeGoogle Scholar
  9. Castaño-Martínez A, Pigueiras G, Sordo MA (2019) On a family of risk measures based on largest claims. Insur Math Econ 86:92–97MathSciNetCrossRefGoogle Scholar
  10. De la Cal J, Cárcamo J (2006) Stochastic orders and majorization of mean order statistics. J Appl Probab 43:704–712MathSciNetCrossRefGoogle Scholar
  11. Donaldson D, Weymark JA (1983) Ethically flexible Gini indices for income distributions in the continuum. J Econ Theory 29:353–358CrossRefGoogle Scholar
  12. Greselin F, Zitikis R (2018) From the classical Gini index of income inequality to a new Zenga-type relative measure of risk: a modeller’s perspective. Econometrics 6:1–20CrossRefGoogle Scholar
  13. Joe H (1997) Multivariate models and multivariate dependence concepts. Chapman and Hall/CRC, Boca RatonCrossRefGoogle Scholar
  14. Kakwani N (1980) On a class of poverty measures. Econometrica 48:437–446MathSciNetCrossRefGoogle Scholar
  15. Karlin S, Rinott Y (1980) Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions. J Multivariate Anal 10:467–498MathSciNetCrossRefGoogle Scholar
  16. Li C, Li X (2018) Preservation of increasing convex/concave order under the formation of parallel/series system of dependent components. Metrika 81:445–464MathSciNetCrossRefGoogle Scholar
  17. Müller A, Scarsini M (2001) Stochastic comparison of random vectors with a common copula. Math Oper Res 26:723–740MathSciNetCrossRefGoogle Scholar
  18. Nelsen RB (2006) An introduction to copulas, 2nd edn. Springer, New YorkzbMATHGoogle Scholar
  19. Pakes A (2004) Convergence and divergence of random series. Aust N Z J Stat 46:29–40MathSciNetCrossRefGoogle Scholar
  20. Pearson K (1934) Tables of the incomplete beta function. Cambridge University Press, CambridgezbMATHGoogle Scholar
  21. Ramos HM, Ollero J, Sordo MA (2000) A sufficient condition for generalized Lorenz order. J Econ Theory 90:286–292MathSciNetCrossRefGoogle Scholar
  22. Rao CR (1965) On discrete distributions arising out of methods of ascertainment. Sankhyā Ser A 27:311–324MathSciNetzbMATHGoogle Scholar
  23. Shaked M, Shanthikumar JG (2007) Stochastic Orders. In: Series: Springer series in statistics. Springer, BerlinGoogle Scholar
  24. Shorrocks AF (1983) Ranking income distributions. Economica 50:3–17CrossRefGoogle Scholar
  25. Son HH (2011) Equity and well-being: measurement and policy practice. Taylor & Francis Group, LondonGoogle Scholar
  26. Tarsitano A (1990) The Bonferroni index of income inequality. In: Dagum C, Zenga M (eds) Income and wealth distribution. inequality and poverty. Springer, Berlin, pp 228–242CrossRefGoogle Scholar
  27. Weymark JA (1981) Generalized Gini inequality indices. Math Soc Sci 1:409–430MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Statistics and Operations ResearchUniversity of CádizCádizSpain
  2. 2.Department of Statistics and Insurance ScienceUniversity of PiraeusPiraeusGreece

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