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CHEBYSHEV INEQUALITIES WITH LAW-INVARIANT DEVIATION MEASURES

Published online by Cambridge University Press:  21 December 2009

Bogdan Grechuk ,
Anton Molyboha  and
Michael Zabarankin
Bogdan Grechuk
Affiliation:
Department of Mathematical Sciences, Stevens Institute of Technology, Castle Point on Hudson, Hoboken, NJ 07030 E-mail: bgrechuk@stevens.edu; amolyboh@stevens.edu; mzabaran@stevens.edu
Anton Molyboha
Affiliation:
Department of Mathematical Sciences, Stevens Institute of Technology, Castle Point on Hudson, Hoboken, NJ 07030 E-mail: bgrechuk@stevens.edu; amolyboh@stevens.edu; mzabaran@stevens.edu
Michael Zabarankin
Affiliation:
Department of Mathematical Sciences, Stevens Institute of Technology, Castle Point on Hudson, Hoboken, NJ 07030 E-mail: bgrechuk@stevens.edu; amolyboh@stevens.edu; mzabaran@stevens.edu
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Abstract

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The consistency of law-invariant general deviation measures with concave ordering has been used to generalize the Rao–Blackwell theorem and to develop an approach for reducing minimization of law-invariant deviation measures to minimization of the measures on subsets of undominated random variables with respect to concave ordering. This approach has been applied for constructing the Chebyshev and Kolmogorov inequalities with law-invariant deviation measures—in particular with mean absolute deviation, lower semideviation and conditional value-at-risk deviation. Additionally, an advantage of the Kolmogorov inequality with certain deviation measures has been illustrated in estimating the probability of the exchange rate of two currencies to be within specified bounds.

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 24 , Issue 1 , January 2010 , pp. 145 - 170
Copyright
Copyright © Cambridge University Press 2009

References

1.Bergstrom, T. & Bagnoli, M. (2005). Log-concave probability and its applications. Economic Theory 26: 445–469.Google Scholar
2.Dana, R.-A. (2005). A representation result for concave Schur-concave functions. Mathematical Finance 15(4): 613–634.Google Scholar
3.Follmer, H. & Schied, A. (2004). Stochastic finance, 2nd ed. Berlin: de Gruyter.Google Scholar
4.Hanoch, G. & Levy, H. (1969). The efficiency analysis of choices involving risk. Review of Economic Studies 36: 335–346.Google Scholar
5.Hogg, R.V., Craig, A., & McKean, J.W. (2004). Introduction to mathematical statistics, 6th ed. New York: Prentice Hall.Google Scholar
6.Kantorovich, L.V. & Akilov, G.P. (1964). Variational methods for the study of nonlinear operators. San Francisco: Holden-Day.Google Scholar
7.Kurdila, A. & Zabarankin, M. (2005). Convex functional analysis. Systems and Control: Foundations and Applications. Basel: Birkhauser.Google Scholar
8.Levy, H. (1998). Stochastic dominance. Boston: Kluwer Academic.Google Scholar
9.Markowitz, H.M. (1952). Portfolio selection. Journal of Finance 7(1): 77–91.Google Scholar
10.Rockafellar, R.T., Uryasev, S. & Zabarankin, M. (2002). Deviation measures in risk analysis and optimization. Report 2002-7, ISE Department, University of Florida, Tampa.Google Scholar
11.Rockafellar, R.T., Uryasev, S. & Zabarankin, M. (2006). Generalized deviations in risk analysis. Finance and Stochastics 10(1): 51–74.Google Scholar
12.Rockafellar, R.T., Uryasev, S. & Zabarankin, M. (2006). Master funds in portfolio analysis with general deviation measures. Journal of Banking and Finance 30(2): 743–77.Google Scholar
13.Rockafellar, R.T., Uryasev, S. & Zabarankin, M. (2006). Optimality conditions in portfolio analysis with general deviation measures. Mathematical Programming 108 (2–3): 515–540.Google Scholar
14.Rockafellar, R.T., Uryasev, S. & Zabarankin, M. (2007). Equilibrium with investors using a diversity of deviation measures. Journal of Banking and Finance 31(11): 3251–3268.Google Scholar
15.Rockafellar, R.T., Uryasev, S., & Zabarankin, M. (2008). Risk tuning with generalized linear regression. Mathematics of Operations Research 33(3): 712–729.CrossRefGoogle Scholar
16.Roy, A. (1952). Safety first and the holding of assets. Econometrica 20: 431–449.Google Scholar
17.Smith, J. (1995). Generalized Chebyshev inequalities: Theory and applications in decision analysis. Operations Research 43(5): 807–825.Google Scholar