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Analytic Expressions for Multivariate Lorenz Surfaces

Author

Listed:
  • Barry C. Arnold

    (University of California)

  • José María Sarabia

    (University of Cantabria)

Abstract

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.

Suggested Citation

  • Barry C. Arnold & José María Sarabia, 2018. "Analytic Expressions for Multivariate Lorenz Surfaces," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(1), pages 84-111, December.
  • Handle: RePEc:spr:sankha:v:80:y:2018:i:1:d:10.1007_s13171-018-00158-9
    DOI: 10.1007/s13171-018-00158-9
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    References listed on IDEAS

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    1. Sarabia, J. -M. & Castillo, Enrique & Slottje, Daniel J., 1999. "An ordered family of Lorenz curves," Journal of Econometrics, Elsevier, vol. 91(1), pages 43-60, July.
    2. Edward W. (Jed) Frees & Glenn Meyers & A. David Cummings, 2014. "Insurance Ratemaking and a Gini Index," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 81(2), pages 335-366, June.
    3. Duangkamon Chotikapanich & D. S. Prasada Rao & Kam Ki Tang, 2007. "Estimating Income Inequality In China Using Grouped Data And The Generalized Beta Distribution," Review of Income and Wealth, International Association for Research in Income and Wealth, vol. 53(1), pages 127-147, March.
    4. Sarabia, José María & Castillo, Enrique & Pascual, Marta & Sarabia, María, 2005. "Mixture Lorenz curves," Economics Letters, Elsevier, vol. 89(1), pages 89-94, October.
    5. Koshevoy, G. A. & Mosler, K., 1997. "Multivariate Gini Indices," Journal of Multivariate Analysis, Elsevier, vol. 60(2), pages 252-276, February.
    6. James B. McDonald, 2008. "Some Generalized Functions for the Size Distribution of Income," Economic Studies in Inequality, Social Exclusion, and Well-Being, in: Duangkamon Chotikapanich (ed.), Modeling Income Distributions and Lorenz Curves, chapter 3, pages 37-55, Springer.
    7. Taguchi, Tokio, 1988. "On the structure of multivariate concentration -- some relationships among the concentration surface and two variate mean difference and regressions," Computational Statistics & Data Analysis, Elsevier, vol. 6(4), pages 307-334, June.
    8. Frees, Edward W. & Meyers, Glenn & Cummings, A. David, 2011. "Summarizing Insurance Scores Using a Gini Index," Journal of the American Statistical Association, American Statistical Association, vol. 106(495), pages 1085-1098.
    9. Samanthi, Ranadeera Gamage Madhuka & Wei, Wei & Brazauskas, Vytaras, 2016. "Ordering Gini indexes of multivariate elliptical risks," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 84-91.
    10. Gupta, Manash Ranjan, 1984. "Functional Form for Estimating the Lorenz Curve," Econometrica, Econometric Society, vol. 52(5), pages 1313-1314, September.
    11. Guillén, Montserrat & Sarabia, José María & Prieto, Faustino, 2013. "Simple risk measure calculations for sums of positive random variables," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 273-280.
    12. Sarabia, José María & Gómez-Déniz, Emilio & Sarabia, María & Prieto, Faustino, 2010. "A general method for generating parametric Lorenz and Leimkuhler curves," Journal of Informetrics, Elsevier, vol. 4(4), pages 524-539.
    13. Yang, Xipei & Frees, Edward W. & Zhang, Zhengjun, 2011. "A generalized beta copula with applications in modeling multivariate long-tailed data," Insurance: Mathematics and Economics, Elsevier, vol. 49(2), pages 265-284, September.
    14. Devroye, Luc, 1993. "A triptych of discrete distributions related to the stable law," Statistics & Probability Letters, Elsevier, vol. 18(5), pages 349-351, December.
    15. Gastwirth, Joseph L, 1971. "A General Definition of the Lorenz Curve," Econometrica, Econometric Society, vol. 39(6), pages 1037-1039, November.
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