Abstract
In this note we establish some appropriate conditions for stochastic equality of two random variables/ vectors which are ordered with respect to convex ordering or with respect to supermodular ordering. Multivariate extensions of this result are also considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cheung. K.C. Characterizing a comonotonic random vector by the distribution of the sum of its components. Insurance: Mathematics and Economics, 47(2): 130–136 (2010)
Cheung, K.C., Dhaene, J., Kukush, A., Linders, D. Ordered random vectors and equality in distribution. Scandinavian Actuarial Journal, 2015(3): 221–244 (2015)
Denuit, M., Dhaene, J., Goovaerts, M.J., Kaas, R. Actuarial Theory for Dependent Risks: Measures, Orders and Models. John Wiley & Sons, Chichester, 2005
Dhaene, J., Denuit, M., Goovaerts, M., Kaas, R., Vyncke, D. The concept of comonotonicity in actuarial science and finance: theory. Insurance: Mathematics and Economics, 31(1): 3–33 (2002)
Dhaene, J., Kukush, A., Linders, D., Tang, Q. Remarks on quantiles and distortion risk measures. European Actuarial Journal, 2: 319–328 (2012)
Dhaene, J., Vanduffel, S., Goovaerts, M.J., Kaas, R., Tang, Q., Vyncke, D. Risk measures and comonotonicity: a review. Stochastic Models, 22, 573–606 (2006)
Fölmer, H., Schied, A. Stochastic Finance. Walter de Gruyter & Co., Berlin, 2004
Hewitt, E., Stromberg, K. Real and Abstract Analysis. Springer Verlag, Berlin, 1965
Joe, H. Multivariate Models and Dependence Concepts. Chapman & Hall, London, 1997
Kaas, R., Dhaene, J., Vyncke, D., Goovaerts, M., Denuit, M. A simple geometry proof that comonotonic risks have the convex-largest sum. ASTIN Bulletin, 32: 71–80 (2002)
Mao, T., Hu, T. A new proof of Cheung’s characterization of comonotonicity. Insurance: Mathematics and Economics, 48(2): 214–216 (2011)
Niculescu, C.P., Persson, L.E. Convex Functions and Their Applications: a Contemporary Approach. Springer, New York, 2006
Shanthikumar, J.G., Shaked, M. Stochastic Orders. Springer, New York, 2007
Zhu, D., Yin, C.C. Two sufficient conditions for convex ordering on risk aggregation. Abstract and Applied Analysis, Volume 2018, Article ID 2937895, 5 pages https://doi.org/10.1155/2018/2937895
Acknowledgements
The authors are grateful to the anonymous referee’s careful reading and detailed helpful comments and constructive suggestions, which have led to a significant improvement of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research was supported by the National Natural Science Foundation of China (11571198, 11701319).
Rights and permissions
About this article
Cite this article
Yin, Cc. Remarks on Equality of Two Distributions under Some Partial Orders. Acta Math. Appl. Sin. Engl. Ser. 34, 274–280 (2018). https://doi.org/10.1007/s10255-018-0744-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10255-018-0744-z