Lorenz map, inequality ordering and curves based on multidimensional rearrangements (2203.09000v4)
Abstract: We propose a multivariate extension of the Lorenz curve based on multivariate rearrangements of optimal transport theory. We define a vector Lorenz map as the integral of the vector quantile map associated with a multivariate resource allocation. Each component of the Lorenz map is the cumulative share of each resource, as in the traditional univariate case. The pointwise ordering of such Lorenz maps defines a new multivariate majorization order, which is equivalent to preference by any social planner with inequality averse multivariate rank dependent social evaluation functional. We define a family of multi-attribute Gini index and complete ordering based on the Lorenz map. We propose the level sets of an Inverse Lorenz Function as a practical tool to visualize and compare inequality in two dimensions, and apply it to income-wealth inequality in the United States between 1989 and 2022.
- R. Aaberge and A. Brandolini. Multidimensional poverty and inequality. Discussion Papers, No. 792, Statistics Norway, Research Department, Oslo, 2014.
- F. Andreoli and C. Zoli. From unidimensional to multidimensional inequality: a review. Metron, 78:5–42, 2020.
- B. Arnold. The Lorenz curve: Evergreen after 110 years. In Advances on Income Inequality and Concentration Measures. Routledge, 2008.
- B. C. Arnold. Pareto Distributions. International Co-operative Publishing House, 1983.
- B. C. Arnold. Majorization and the Lorenz order: A brief introduction. Springer Science & Business Media, 2012.
- Majorization and the Lorenz order with Applications in Applied Mathematics and Economics. Springer, 2018.
- A. Atkinson and F. Bourguignon. The comparison of multi-dimensioned distributions of economic status. Review of Economic Studies, 49:183–201, 1982.
- Minkowski-type theorems and least-squares clustering. Algorithmica, 20:61–76, 1998.
- A. Banerjee. Multidimensional Gini index. Mathematical Social Sciences, 60:87–93, 2010.
- A. Banerjee. Multidimensional Lorenz dominance: a definition and an example. Keio Economic Studies, 52:65–80, 2016.
- P. Bickel and E. Lehmann. Descriptive statistics for nonparametric models, iii. dispersion. Annals of Statistics, 4:1139–1158, 1976.
- S. Bonhomme and J.-M. Robin. Assessing the equalizing force of mobility using short panels: France, 1990–2000. Review of Economic Studies, 76:63–92, 2009.
- Y. Brenier. Polar factorization and monotone rearrangement of vector‐valued functions. Communications on Pure and Applied Mathematics, 4:375–417, 1991.
- Changes in US family finances from 2013 to 2016: Evidence from the survey of consumer finances. Fed. Res. Bull., 103:1, 2017.
- W. Brock and R. Thomson. Convex solutions of implicit relations. Mathematics Magazine, 39:208–111, 1966.
- From knothe’s transport to brenier’s map and a continuation method for optimal transport. SIAM Journal on Mathematical Analysis, 41:2554–2576, 2010.
- Local utility and multivariate risk aversion. Mathematics of Operations Research, 41:266–276, 2016.
- Monge-Kantorovich depth, quantiles, ranks and signs. Annals of Statistics, 45:223–256, 2017.
- Dardanoni. Measuring social mobility. Journal of Economic Theory, 61:372–394, 1993.
- R. Davidson and J.-Y. Duclos. Statistical inference for stochastic dominance and for the measurement of poverty and inequality. Econometrica, 68:1435–1464, 2000.
- N. Deb and B. Sen. Multivariate rank-based distribution-free nonparametric testing using measure transportation. Journal of the American Statistical Association, 118:192–207, 2023.
- K. Decancq and M. Lugo. Inequality of wellbeing: A multidimensional approach. Economica, 79:721–746, 2012.
- R. Dudley. Real Analysis and Probability. Cambridge University Press, 2002.
- Comonotone measures of multivariate risks. Mathematical Finance, 22:109–132, 2012.
- O. Faugeras and Rüschendorf. Markov morphisms: a combined copula and mass transportation approach to multivariate quantiles. Mathematica Applicanda, 45:3–45, 2017.
- M. Faure and N. Gravel. Reducing inequalities among unequals. International Economic Review, 62:357–404, 2021.
- A. Figalli. On the continuity of center-outward distribution and quantile functions. Nonlinear Analysis, 177:413–429, 2018.
- F. Fisher. Income distribution, value judgments and welfare. Quarterly Journal of Economics, 70:380–424, 1956.
- T. Gajdos and J. Weymark. Multidimensional generalized Gini indices. Economic Theory, 26:471–496, 2005.
- A. Galichon and M. Henry. Dual theory of choice under multivariate risks. Journal of Economic Theory, 147:1501–1516, 2012.
- J. L. Gastwirth. A general definition of the Lorenz curve. Econometrica, 39:1037–1039, 1971.
- P. Ghosal and B. Sen. Multivariate ranks and quantiles using optimal transport: Consistency, rates, and nonparametric testing. Annals of Statistics, 2022.
- N. Gravel and P. Moyes. Ethically robust comparisons of bi-dimensionnal distributions with an ordinal attribute. Journal of Economic Theory, 147:1384–1426, 2012.
- A multivariate extension of the Lorenz curve based on copulas and a related multivariate gini coefficient. Journal of Economic Inequality, 20:727–748, 2022.
- M. Hallin. Measure transportation and statistical decision theory. Annual Review of Statistics and Its Application, 9:410–424, 2022.
- M. Hallin and G. Mordant. Center-outward multiple-output lorenz curves and gini indices a measure transportation approach. Unpublished manuscript, 2022.
- Behind the numbers: Understanding the survey of consumer finances. Journal of Financial Counseling and Planning, 29:410–418, 2018.
- Inequalities. Cambridge University Press, 1934.
- H. Joe. Multivariate Models and Multivariate Dependence Concepts. Chapman and Hall, 1997.
- A. Kennickell. Multiple imputation in the survey of consumer finances. Statistical Journal of the IAOS, 33:143–151, 2017a.
- A. B. Kennickell. The bitter end? the close of the 2007 SCF field period. Statistical Journal of the IAOS, 33:93–99, 2017b.
- Consistent weight design for the 1989, 1992 and 1995 SCFs, and the distribution of wealth. Review of Income and Wealth, 45:193–215, 1999.
- Convergence of a newton algorithm for semi-discrete optimal transport. Journal of the European Mathematical Society, 21:2603–2651, 2019.
- H. Knothe. Contributions to the theory of convex bodies. Michigan Mathematical Journal, 4:39–52, 1957.
- S.-C. Kolm. Multidimensional egalitarianisms. The Quarterly Journal of Economics, 91:1–13, 1977.
- G. Koshevoy and K. Mosler. The Lorenz zonoid of a multivariate distribution. Journal of the American Statistical Association, 91:873–882, 1996.
- G. Koshevoy and K. Mosler. Multivariate Gini indices. Journal of Multivariate Analysis, 60:252–276, 1997.
- G. Koshevoy and K. Mosler. Price majorization and the inverse Lorenz function. Discussion Papers in Statistics and Econometrics 3/99, University of Cologne, 1999.
- G. Koshevoy and K. Mosler. Multivariate Lorenz dominance based on zonoids. AStA, 91:57–76, 2007.
- E. L. Lehmann. Some concepts of dependence. The Annals of Mathematical Statistics, 37:1137–1153, 1966.
- B. Lévy. A numerical algorithm for L2 semi-discrete optimal transport in 3D. ESAIM: Mathematical Modelling and Numerical Analysis, 49:1693–1715, 2015.
- Statistical analysis with missing data, volume 793. John Wiley & Sons, 2019.
- M. Lorenz. Methods of measuring the concentration of wealth. Publication of the American Statistical Association, 9:209–219, 1905.
- Inequalities: Theory of Majorization and Its Applications. Springer, 2nd edition, 2011.
- R. J. McCann. Existence and uniqueness of monotone measure-preserving maps. Duke Mathematical Journal, 80:309–323, 1995.
- A. Müller and D. Stoyan. Comparison Methods for Stochastic Models and Risks. Wiley, 2002.
- G. Peyré and M. Cuturi. Computational Optimal Transport. ArXiv:1803.00567, 2018.
- G. Puccetti and M. Scarsini. Multivariate comonotonicity. Journal of Multivariate Analysis, 101:291–304, 2010.
- J. Quiggin. Increasing risk: another definition. In A. Chikan, editor, Progress in Decision, Utility and Risk Theory, pages 239–248. Kluwer: Dordrecht, 1992.
- S. Rachev and L. Rüschendorf. A characterization of random variables with minimal L2 distance. Journal of Multivariate Analysis, 32:48–54, 1990.
- R. T. Rockafellar. Characterization of the subdifferentials of convex functions. Pacific Journal of Mathematics, 17:497–510, 1966.
- M. Rosenblatt. Remarks on a multivariate transformation. Annals of Mathematical Statistics, 23:470–472, 1952.
- D. B. Rubin. Multiple imputation after 18+ years. Journal of the American Statistical Association, 91:473–489, 1996.
- J.-M. Sarabia and V. Jorda. Bivariate Lorenz curves: a review of recent proposals. In XXII Jornadas de ASEPUMA, 2014.
- J.-M. Sarabia and V. Jorda. Lorenz surfaces based on the Sarmanov–Lee distribution with applications to multidimensional inequality in well-being. Mathematics, 8:1–17, 2020.
- E. Savaglio. Three approaches to the analysis of multidimensional inequality, chapter 10. Routledge, 2006.
- M. Shaked and J. G. Shanthikumar. Stochastic Orders. Springer, 2007.
- Distribution-free consistent independence tests via center-outward ranks and signs. Journal of the American Statistical Association, 117:395–410, 2022.
- A. F. Shorrocks. Ranking income distributions. Economica, 197:3–17, 1983.
- T. Taguchi. On the two-dimensional concentration surface and extensions of concentration coefficient and Pareto distribution to the two-dimensional case-i. Annals of the Institute of Statistical Mathematics, 24:355–382, 1972a.
- T. Taguchi. On the two-dimensional concentration surface and extensions of concentration coefficient and Pareto distribution to the two-dimensional case-ii. Annals of the Institute of Statistical Mathematics, 24:599–619, 1972b.
- C. Villani. Topics in Optimal Tranportation. American Mathematical Society, 2003.
- C. Villani. Optimal Transport: Old and New. Springer, 2009.
- J. Weymark. Generalized Gini inequality indices. Mathematical Social Sciences, 1:409–430, 1981.
- E. N. Wolff. Household wealth trends in the United States, 1962 to 2019: Median wealth rebounds… but not enough. Working paper, National Bureau of Economic Research, 2021.
- M. E. Yaari. The dual theory of choice under risk. Econometrica, 55:95–115, 1987.