Sankhya A

pp 1–28 | Cite as

Analytic Expressions for Multivariate Lorenz Surfaces

  • Barry C. ArnoldEmail author
  • José María Sarabia


The Lorenz curve is a much used instrument in economic analysis. It is typically used for measuring inequality and concentration. In insurance, it is used to compare the riskiness of portfolios, to order reinsurance contracts and to summarize relativity scores (see Frees et al. J. Am. Statist. Assoc.106, 1085–1098, 2011; J. Risk Insur.81, 335–366, 2014 and Samanthi et al. Insur. Math. Econ.68, 84–91, 2016). It is sometimes called a concentration curve and, with this designation, it attracted the attention of Mahalanobis (Econometrica28, 335–351, 1960) in his well known paper on fractile graphical analysis. The extension of the Lorenz curve to higher dimensions is not a simple task. There are three proposed definitions for a suitable Lorenz surface, proposed by Taguchi (Ann. Inst. Statist. Math.24, 355–382, 1972a, 599–619, 1972b; Comput. Stat. Data Anal.6, 307–334, 1988) and Lunetta (1972), Arnold (1987, 2015) and Koshevoy and Mosler (J. Am. Statist. Assoc.91, 873–882, 1996). In this paper, using the definition proposed by Arnold (1987, 2015), we obtain analytic expressions for many multivariate Lorenz surfaces. We consider two general classes of models. The first is based on mixtures of Lorenz surfaces and the second one is based on some simple classes of bivariate mixture distributions.


Multivariate distributions Mixture distributions Laplace transform Gini index. 

AMS (2000) subject classification

Primary 62E10; Secondary 91B82 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The second author gratefully acknowledge financial support from the Programa Estatal de Fomento de la Investigación Científica y Técnica de Excelencia, Spanish Ministry of Economy and Competitiveness, project ECO2016-76203-C2-1-P. The authors thank two anonymous reviewers for helpful and detailed comments and suggestions that have improved the paper.


  1. Albano, M., Amdeberhan, T., Beyerstedt, E. and Moll, V.H. (2011). The integrals in Gradshteyn and Ryzhik. Part 19: the error function. Scientia, Ser. A: Math. Sci. 21, 25–42.MathSciNetzbMATHGoogle Scholar
  2. Arnold, B.C. (1987). Majorization and the Lorenz Curve. Lecture Notes in Statistics 43. Springer, New York.Google Scholar
  3. Arnold, B.C. (2008). The Lorenz curve: evergreen after 100 years. In Advances on Income Inequality and and Concentration Measures, (G. Betti and A. Lemmi, eds.), pp. 12–24. Routledge, New York.Google Scholar
  4. Arnold, B.C. (2015). Pareto Distributions, 2nd edn. Chapman and Hall/CRC, Boca Raton.Google Scholar
  5. Bailey, W.N. (1935). Appell’s Hypergeometric Functions of Two Variables. Cambridge University Press, Cambridge,.Google Scholar
  6. Balakrishnan, N. and Lai, C.-D. (2009). Continuous Bivariate Distributions. Springer, New York.zbMATHGoogle Scholar
  7. Chotikapanich, D., Rao, D.S.P. and Tang, K.K. (2007). Estimating income inequality in China using grouped data and the generalized beta distribution. Rev. Income Wealth 53, 127–147.CrossRefGoogle Scholar
  8. Cockriel, W.M. and McDonald, J.B. (2017). Two multivariate generalized beta families. Communications in Statistics, Theory and Methods, Forthcoming.Google Scholar
  9. Devroye, L. (1993). A triptych of discrete distributions related to the stable law. Statistics and Probability Letters 18, 349–351.MathSciNetCrossRefzbMATHGoogle Scholar
  10. Denuit, M. and Vermandale, C. (1999). Lorenz and excess wealth orders, with applications in reinsurance theory. Scand. Actuar. J. 2, 170–185.MathSciNetCrossRefzbMATHGoogle Scholar
  11. Frees, E.W., Meyers, G. and Cummings, A.D. (2011). Summarizing insurance scores using a Gini index. J. Am. Statist. Assoc. 106, 1085–1098.MathSciNetCrossRefzbMATHGoogle Scholar
  12. Frees, E.W., Meyers, G. and Cummings, A.D. (2014). Insurance ratemaking and a Gini index. J. Risk Insur. 81, 335–366.CrossRefGoogle Scholar
  13. Gastwirth, J.L. (1971). A general definition of the Lorenz curve. Econometrica 39, 1037–1039.CrossRefzbMATHGoogle Scholar
  14. Genest, C. and MacKay, R.J. (1986). Copules Archimédiennes et families de lois bidimensionnelles dont les marges sont données. Can. J. Stat. 14, 145–159.CrossRefzbMATHGoogle Scholar
  15. Gradshteyn, I.S. and Ryzhik, I.M. (2007). In Table of Integrals, Series, and Products, 7th edn, (A. Jeffrey and D. Zwillinger, eds.). Academic Press, New York.Google Scholar
  16. Guillén, M., Sarabia, J.M. and Prieto, F. (2013). Simple risk measure calculations for sums of positive random variables. Insur. Math. Econ. 53, 273–280.MathSciNetCrossRefzbMATHGoogle Scholar
  17. Gupta, M.R. (1984). Functional form for estimating the Lorenz curve. Econometrica 52, 1313–1314.MathSciNetCrossRefzbMATHGoogle Scholar
  18. Hanley, W.G. (1998). On the Lorenz zonoid representation of distributional variability. Thesis (Ph.D.) University of California, Riverside.Google Scholar
  19. Koshevoy, G. (1995). Multivariate Lorenz majorization. Soc. Choice Welf. 12, 93–102.MathSciNetCrossRefzbMATHGoogle Scholar
  20. Koshevoy, G. and Mosler, K. (1996). The Lorenz zonoid of a multivariate distribution. J. Am. Statist. Assoc. 91, 873–882.MathSciNetCrossRefzbMATHGoogle Scholar
  21. Koshevoy, G. and Mosler, K. (1997). Multivariate Gini indices. J. Multivar. Anal. 60, 252–276.MathSciNetCrossRefzbMATHGoogle Scholar
  22. Lee, M.L.T. (1996). Properties of the Sarmanov family of bivariate distributions. Commun. Stat. Theory Methods 25, 1207–1222.MathSciNetCrossRefzbMATHGoogle Scholar
  23. Lunetta, G. (1972). Di un indice di cocentrazione per variabili statistische doppie. Annali della Facoltá di Economia e Commercio dell Universitá di Catania A 18.Google Scholar
  24. Mahalanobis, P.C. (1960). A method of fractile graphical analysis. Econometrica 28, 335–351.CrossRefGoogle Scholar
  25. Mahalanobis, P.C. (1970). Extensions of fractile graphical analysis to higher dimensional data. University of North Carolina Press, Chapel Hill, p. 397–406.Google Scholar
  26. Marshall, A.W. and Olkin, I. (1988). Families of multivariate distributions. J. Am. Statist. Assoc. 83, 834–841.MathSciNetCrossRefzbMATHGoogle Scholar
  27. Marshall, A. W., Olkin, I. and Arnold, B. C. (2011). Inequalities: theory of majorization and its applications, 2nd edn. Springer, New York.CrossRefzbMATHGoogle Scholar
  28. McDonald, J.B. (1984). Some generalized functions for the size distribution of income. Econometrica 52, 647–663.CrossRefzbMATHGoogle Scholar
  29. Mosler, K. (2002). Multivariate dispersion, central regions and depth: the lift zonoid approach. Lecture Notes Statistics, vol. 165. Springer, Berlin.CrossRefzbMATHGoogle Scholar
  30. Olkin, I. and Liu, R. (2003). A bivariate Lorenz curve. Statist. Probab. Lett. 62, 407–412.MathSciNetCrossRefzbMATHGoogle Scholar
  31. Samanthi, R.G.M., Wei, W. and Brazauskas, V. (2016). Ordering Gini indexes of multivariate elliptical risks. Insur. Math. Econ. 68, 84–91.MathSciNetCrossRefzbMATHGoogle Scholar
  32. Sarabia, J.M. and Jordá, V. (2013). Modeling Bivariate Lorenz Curves with Applications to Multidimensional Inequality in Well-Being. Fifth Meeting of ECINEQ at Bari, Italy.Google Scholar
  33. Sarabia, J.M. and Jordá, V. (2014). Bivariate Lorenz Curves based on the Sarmanov-Lee Distribution. Topics in Statistical Simulation, Springer Proceedings in Mathematics & Statistics Volume 114, 447-455, Springer.Google Scholar
  34. Sarabia, J.M., Castillo, E. and Slottje, D.J. (1999). An ordered family of Lorenz curves. J. Econ. 91, 43–60.MathSciNetCrossRefzbMATHGoogle Scholar
  35. Sarabia, J.M., Castillo, E., Pascual, M. and Sarabia, M. (2005). Mixture Lorenz curves. Econ. Lett. 89, 89–94.MathSciNetCrossRefzbMATHGoogle Scholar
  36. Sarabia, J.M., Gómez-Déniz, E., Sarabia, M. and Prieto, F. (2010). A general method for generating parametric Lorenz and Leimkuhler curves. J. Informetrics 4, 524–539.CrossRefGoogle Scholar
  37. Sarabia, J.M., Prieto, F. and Jordá, V. (2014). Bivariate beta-generated distributions with applications to well-being data. J. Stat. Distrib. Appl. 2014, 1–15.zbMATHGoogle Scholar
  38. Sarmanov, O.V. (1966). Generalized normal correlation and two-dimensional Frechet classes. Doklady Sov. Math. 168, 596–599.MathSciNetzbMATHGoogle Scholar
  39. Sibuya, M. (1979). Generalized hypergeometric, digamma and trigamma distributions. Ann. Inst. Stat. Math. 31, 373–390.MathSciNetCrossRefzbMATHGoogle Scholar
  40. Steutel, F.W. and Van Harn, K. (1979). Discrete analogous of self-decomposability and stability. Ann. Probab. 7, 893–899.MathSciNetCrossRefzbMATHGoogle Scholar
  41. Taguchi, T. (1972a). On the two-dimensional concentration surface and extensions of concentration coefficient and Pareto distribution to the two-dimensional case-I. Ann. Inst. Statist. Math. 24, 355–382.MathSciNetCrossRefzbMATHGoogle Scholar
  42. Taguchi, T. (1972b). On the two-dimensional concentration surface and extensions of concentration coefficient and Pareto distribution to the two-dimensional case-II. Ann. Inst. Statist. Math. 24, 599–619.MathSciNetCrossRefzbMATHGoogle Scholar
  43. Taguchi, T. (1988). On the structure of multivariate concentration-some relationships among the concentration surface and two variate mean difference and regressions. Comput. Stat. Data Anal. 6, 307–334.MathSciNetCrossRefzbMATHGoogle Scholar
  44. Yang, X., Frees, E.W. and Zhang, Z. (2011). A generalized beta copula with applications in modeling multivariate long-tailed data. Insur.: Math. Econ. 49, 265–284.MathSciNetzbMATHGoogle Scholar

Copyright information

© Indian Statistical Institute 2018

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of CaliforniaRiversideUSA
  2. 2.Department of EconomicsUniversity of CantabriaSantanderSpain

Personalised recommendations