Abstract
The temptation to seek multivariate generalizations of majorization and the Lorenz order is strong, and has not been resisted. In an income setting it is reasonable to consider income from several sources or income in incommensurable units. In fact, the idea that income can be measured undimensionally is perhaps the radical point of view, and interest should center on multivariate measures of income. Let us first consider various possible multivariate generalizations of majorization.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Arnold, B. C. (1987). Majorization and the Lorenz order: A brief introduction. Lecture notes in statistics (Vol. 43). Berlin: Springer.
Arnold, B. C. (2015b). Pareto distributions (2nd ed.). Boca Raton, FL: CRC Press, Taylor & Francis Group.
Arnold, B. C., & Sarabia, J. M. (2018). Analytic expressions for multivariate Lorenz surfaces (submitted)
Bairamov, I., & Kotz, S. (2003). On a new family of positive quadrant dependent bivariate distributions. International Journal of Mathematics, 3, 1247–1254.
Darbellay, G. A., & Vajda, I. (2000). Entropy expressions for multivariate continuous distributions. IEEE Transactions on Information Theory, 46, 709–712.
Elton, J., & Hill, T. P. (1992). Fusions of a probability distribution. Annals of Probability, 20, 421–454.
Huang, J. S., & Kotz, S. (1999). Modifications of the Farlie-Gumbel-Morgenstern distributions a tough hill to climb. Metrika, 49, 135–145.
Joe, H., & Verducci, J. (1992). Multivariate majorization by positive combinations. In M. Shaked & Y. L. Tong (Eds.), Stochastic inequalities. IMS lecture notes - Monograph series (Vol. 22, pp. 159–181).
Kakwani, N. C. (1977). Applications of Lorenz curves in economic analysis. Econometrica, 45, 719–728.
Karlin, S., & Rinott, Y. (1988). A generalized Cauchy-Binet formula and applications to total positivity and majorization. Journal of Multivariate Analysis, 27, 284–299.
Koshevoy, G. (1995). Multivariate Lorenz majorization. Social Choice and Welfare, 12, 93–102.
Koshevoy, G., & Mosler, K. (1996). The Lorenz zonoid of a multivariate distribution. Journal of the American Statistical Association, 91, 873–882.
Koshevoy, G., & Mosler, K. (1997). Multivariate Gini indices. Journal of Multivariate Analysis, 60, 252–276.
Lee, M.-L. T. (1996). Properties and applications of the Sarmanov family of bivariate distributions. Communications in Statistics, Theory and Methods, 25, 1207–1222.
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.
Marshall, A. W., Olkin, I., & Arnold, B. C. (2011). Inequalities: Theory of majorization and its applications (2nd ed.). New York: Springer.
Meyer, P. A. (1966). Probability and potentials. Waltham, MA: Blaisdell.
Mosler, K. (2002). Multivariate dispersion, central regions and depth: The lift Zonoid approach. Lecture notes in statistics (Vol. 165). Berlin: Springer.
Sarabia, J. M., & Jordá, V. (2013). Modeling Bivariate Lorenz Curves with Applications to Multidimensional Inequality in Well-Being. Fifth ECINEQ Meeting, Bari, Italy, 201. Document available at: http://www.ecineq.org/ecineq_bari13/documents/booklet05.pdf
Sarabia, J. M., & Jordá, V. (2014a). Bivariate Lorenz curves based on the Sarmanov-Lee distribution. In V. B. Velas, S. Mignani, P. Monari, & L. Salmano (Eds.), Topics in statistical simulation. New York: Springer.
Sarabia, J. M. , & Jordá, V. (2014b). Explicit expressions of the Pietra index for the generalized function for the size distribution of income. Physica A: Statistical Mechanics and Its Applications, 416, 582–595.
Sarabia, J. M., Jordá, V., & Remuzgo, L. (2017a). The Theil indices in parametric families of income distributions - A short review. The Review of Income and Wealth, 63, 867–880.
Taguchi, T. (1972a). On the two-dimensional concentration surface and extensions of concentration coefficient and Pareto distribution to the two dimensional case-I. Annals of Institute of Statistical Mathematics, 24, 355–382.
Taguchi, T. (1972b). On the two-dimensional concentration surface and extensions of concentration coefficient and Pareto distribution to the two dimensional case-II. Annals of Institute of Statistical Mathematics, 24, 599–619.
Whitt, W. (1980). The effect of variability in the GI/G/s queue. Journal of Applied Probability, 17, 1062–1071.
Zografos, K., & Nadarajah, S. (2005). Expressions for Renyi and Shannon entropies for multivariate distributions. Statistics & Probability Letter, 71, 71–84.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Arnold, B.C., Sarabia, J.M. (2018). Multivariate Majorization and Multivariate Lorenz Ordering. In: Majorization and the Lorenz Order with Applications in Applied Mathematics and Economics. Statistics for Social and Behavioral Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-93773-1_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-93773-1_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-93772-4
Online ISBN: 978-3-319-93773-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)