Analytic Expressions for Multivariate Lorenz Surfaces
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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.
KeywordsMultivariate distributions Mixture distributions Laplace transform Gini index.
AMS (2000) subject classificationPrimary 62E10; Secondary 91B82
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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.
- Arnold, B.C. (1987). Majorization and the Lorenz Curve. Lecture Notes in Statistics 43. Springer, New York.Google Scholar
- 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
- Arnold, B.C. (2015). Pareto Distributions, 2nd edn. Chapman and Hall/CRC, Boca Raton.Google Scholar
- Bailey, W.N. (1935). Appell’s Hypergeometric Functions of Two Variables. Cambridge University Press, Cambridge,.Google Scholar
- Cockriel, W.M. and McDonald, J.B. (2017). Two multivariate generalized beta families. Communications in Statistics, Theory and Methods, Forthcoming.Google Scholar
- 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
- Hanley, W.G. (1998). On the Lorenz zonoid representation of distributional variability. Thesis (Ph.D.) University of California, Riverside.Google Scholar
- 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
- 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
- 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
- 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