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Inverse Stochastic Dominance, Majorization, and Mean Order Statistics

Part of: Inequalities Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Jesús De La Cal  and
Javier Cárcamo
Jesús De La Cal*
Affiliation:
Universidad del País Vasco
Javier Cárcamo*
Affiliation:
Universidad Autónoma de Madrid
*
∗Postal address: Departamento de Matemática Aplicada y Estadística e Investigación Operativa, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain. Email address: jesus.delacal@ehu.es
∗∗Postal address: Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain. Email address: javier.carcamo@uam.es
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Abstract

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The inverse stochastic dominance of degree r is a stochastic order of interest in several branches of economics. We discuss it in depth, the central point being the characterization in terms of the weak r-majorization of the vectors of expected order statistics. The weak r-majorization (a notion introduced in the paper) is a natural extension of the classical (reverse) weak majorization of Hardy, Littlewood and Pòlya. This work also shows the equivalence between the continuous majorization (of higher order) and the discrete r-majorization. In particular, our results make it clear that the cases r = 1, 2 differ substantially from those with r ≥ 3, a fact observed earlier by Muliere and Scarsini (1989), among other authors. Motivated by this fact, we introduce new stochastic orderings, as well as new social inequality indices to compare the distribution of the wealth in two populations, which could be considered as natural extensions of the first two dominance rules and the S-Gini indices, respectively.

Keywords

Stochastic ordersinverse stochastic dominanceorder statisticmajorizationS-Gini indexinequality measurement

MSC classification

Secondary: 60E15: Inequalities; stochastic orderings 60E99: None of the above, but in this section 26D15: Inequalities for sums, series and integrals
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 1 , March 2010 , pp. 277 - 292
Copyright
Copyright © Applied Probability Trust 2010 

References

Aaberge, R. (2009). Ranking intersecting Lorenz curves. Soc. Choice Welf. 33, 235–259.Google Scholar
Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (2008). A First Course in Order Statistics. Society for Industrial and Applied Mathematics, Philadelphia, PA.Google Scholar
Baı´llo, A., Berrendero, J. R. and Cárcamo, J. (2009). Tests for zero-inflation and overdispersion: a new approach based on the stochastic convex order. Comput. Statist. Data Anal. 53, 2628–2639.CrossRefGoogle Scholar
Barrett, G. F. and Donald, S. G. (2009). Statistical inference with generalized Gini indices of inequality, poverty, and welfare. J. Business Econom. Statist. 27, 1–17.Google Scholar
Berrendero, J. R. and Cárcamo, J. (2009). Characterizations of exponentiality within the HNBUE class and related tests. J. Statist. Planning Infer. 139, 2399–2406.Google Scholar
Berrendero, J. R. and Cárcamo, J. (2010). Tests for the second order stochastic dominance based on L-statistics. To appear in J. Business Econom. Statist. Google Scholar
De la Cal, J. and Cárcamo, J. (2006). Stochastic orders and majorization of mean order statistics. J. Appl. Prob. 43, 704–712.Google Scholar
Dentcheva, D. and Ruszczynski, A. (2006). Inverse stochastic dominance constraints and rank dependent expected utility theory. Math. Program. 108, 297–311.Google Scholar
Denuit, M., Lefèvre, C. and Shaked, M. (1998). The s-convex orders among real random variables, with applications. Math. Inequal. Appl. 1, 585–613.Google Scholar
Denuit, M., Lefèvre, C. and Shaked, M. (2000). On the theory of high convexity stochastic orders. Statist. Prob. Lett. 47, 287–293.Google Scholar
Denuit, M., Dhaene, J., Goovaerts, M. and Kaas, R. (2005). Actuarial Theory for Dependent Risks. John Wiley, Chichester.Google Scholar
Donaldson, D. and Weymark, J. A. (1980). A single-parameter generalization of the Gini indices of inequality. J. Econom. Theory 22, 67–86.Google Scholar
Donaldson, D. and Weymark, J. A. (1983). Ethical flexible indices for income distributions in the continuum. J. Econom. Theory 29, 353–358.Google Scholar
Fishburn, P. C. (1976). Continua of stochastic dominance relations for bounded probability distributions. J. Math. Econom. 3, 295–311.Google Scholar
Huang, J. S. (1989). Moment problem of order statistics: a review. Internat. Statist. Rev. 57, 59–66.Google Scholar
Johnson, N. L., Kemp, A. W. and Kotz, S. (2005). Univariate Discrete Distributions, 3rd edn. John Wiley, Hoboken, NJ.Google Scholar
Kakwani, N. (1980). On a class of poverty measures. Econometrica 48, 437–446.Google Scholar
Lai, C.-D. and Xie, M. (2006). Stochastic Ageing and Dependence for Reliability. Springer, Berlin.Google Scholar
Lambert, P. J. (1993). The Distribution and Redistribution of Income, a Mathematical Analysis, 2nd edn. Basil-Blackwell, Oxford.Google Scholar
Levy, H. (1992). Stochastic dominance and expected utility: survey and analysis. Manag. Sci. 38, 555–593.Google Scholar
Maccheroni, F., Muliere, P. and Zoli, C. (2005). Inverse stochastic orders and generalized Gini functionals. Metron 53, 529–559.Google Scholar
Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.Google Scholar
Muliere, P. and Scarsini, M. (1989). A note on stochastic dominance and inequality measures. J. Econom. Theory 49, 314–323.Google Scholar
Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.Google Scholar
Shaked, M. and Shanthikumar, J. G. (2006). Stochastic Orders. Springer, New York.Google Scholar
Shorrocks, A. F. and Foster, J. E. (1987). Transfer sensitive inequality measures. Rev. Econom. Stud. 54, 485–497.Google Scholar
Wang, S. S. and Young, V. R. (1998). Ordering risks: expected utility theory versus Yaari's dual theory of risk. Insurance Math. Econom. 22, 145–161.Google Scholar
Yaari, M. E. (1987). The dual theory of choice under risk. Econometrica 55, 95–115.Google Scholar
Yitzhaki, S. (1983). On the extension of the Gini inequality index. Internat. Econom. Rev. 24, 617–628.Google Scholar
Zoli, C. (1999). Intersecting generalized Lorenz curves and the Gini index. Soc. Choice Welf. 16, 183–196.Google Scholar
Zoli, C. (2002). Inverse stochastic dominance, inequality measurement and Gini indices. In Inequalities: Theory, Measurement and Applications (Bordeaux, 2000), Springer, Vienna, pp. 119–161.Google Scholar