Abstract
It is well-known that the Lorenz ordering, which is widely used to rank the inequality of income, will lead to the ordering of coefficient of variation. This paper finds that these two stochastic orders are equivalent within several common two-parameter families of distributions including the location-scale family, some scale and shape parameter family. Our finding manifests that once the compared life distributions or income distributions belong to a two-parameter family discussed above, rankings by the Lorenz curve and by the coefficient of variation for inequality generate the same order. Furthermore, a simple general sufficient condition without limiting within two-parameter families for this property is provided. These results could extend application of coefficient of variation, which can be regarded as a proxy of Lorenz curve in many cases for an inequality ranking or orderings of life distributions, even if the life has asymmetric heavy-tail distribution.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Atkinson, A.B.: On the measurement of inequality. J. Econ. Theory. 5, 244–263 (1970)
Rothschild, M., Stiglitz, J.E.: Some further results on the measurement of in equality. J. Econ. Theory. 6, 188–204 (1973)
Cowell, F.: Measuring Inequality, 3rd edn. Oxford University Press, New York (2011)
Foster, J.E., Shorrocks, A.F.: Inequality and poverty orderings. Eur. Econ. Rev. 32, 654–661 (1988)
Marshall, A.W., Olkin, I.: Life Distributions. Springer, New York (2007). https://doi.org/10.1007/978-0-387-68477-2
Shaked, M., Shanthikumar, J.G.: Stochastic Orders. Springer, New York (2007). https://doi.org/10.1007/978-0-387-34675-5
Marshall, A.W., Olkin, I., Arnold, B.: Inequalities: Theory of Majorization and Its Applications, 2nd edn. Springer, New York (2010). https://doi.org/10.1007/978-0-387-68276-1
Gini, C.: Variabilità e mutabilità. Studi Economico-Giuridici dell’Università di Cagliari, vol. 3, pp. 1–158 (1912)
Pietra, G.: Delle relazioni fra indici di variabilita, Note I e II. Atti del Reale Instituto Veneto di Scienze, Lettere ed Arti, vol. 74, pp. 775–804 (1915)
Amato, V.: Metodologia Statistica Strutturale. Cacucci, Bari (1968)
Arnold, B.C.: Majorization and the Lorenz Order: A Brief Introduction. Lecture Notes in Statistics, vol. 43. Springer, Heidelberg (1987). https://doi.org/10.1007/978-1-4615-7379-1
Meyer, J.: Two-moment decision models and expected utility maximization. Am. Econ. Rev. 77, 421–430 (1987)
Levy, H.: Two-moment decision models and expected utility maximization: comment. Am. Econ. Rev. 79, 597–600 (1989)
Schuhmacher, F., Eling, M.: A decision-theoretic foundation for reward-to-risk performance measures. J. Bank. Financ. 36, 2077–2082 (2012)
Rinne, H.: Location-Scale Distributions. In: Lovric, M. (ed.) International Encyclopedia of Statistical Science, pp. 752–754. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-04898-2
Newby, M.J.: The properties of moment estimators for the Weibull distribution based on the sample coefficient of variation. Technometrics 22, 187–194 (1980)
Acknowledgments
The authors would like to thank Professor Yichun Chi at the Central University of Finance and Economics, Professor Qihe Tang at the University of Iowa and Professor Xiaojun Shi at the Remnin University of China for helpful comments. The first author was supported by the Major Project of the National Social Science Fund of China (16ZDA052), and by the MOE(China) National Key Research Bases for Humanities and Social Sciences (16JJD910001).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Xiao, Y., Yao, J. (2019). On the Equivalence of the Coefficient of Variation Ordering and the Lorenz Ordering Within Two-Parameter Families. In: Li, QL., Wang, J., Yu, HB. (eds) Stochastic Models in Reliability, Network Security and System Safety. JHC80 2019. Communications in Computer and Information Science, vol 1102. Springer, Singapore. https://doi.org/10.1007/978-981-15-0864-6_14
Download citation
DOI: https://doi.org/10.1007/978-981-15-0864-6_14
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-0863-9
Online ISBN: 978-981-15-0864-6
eBook Packages: Computer ScienceComputer Science (R0)