Title: Analytic Expressions for Multivariate Lorenz Surfaces

Author(s): Barry C. Arnold and Jose Maria Sarabia
Issue: Volume 80 Series A Part 3 Year 2018
Pages: 84 -- 111
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 (Econometrica 28, 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.
Primary 62E10; Secondary 91B82
Keywords and phrases: Multivariate distributions, Mixture distributions, Laplace transform, Gini index.