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More bounds on the expectation of a convex function of a random variable

Published online by Cambridge University Press:  14 July 2016

Ben-Tal A.  and
E. Hochman
Ben-Tal A.*
Affiliation:
University of Tel-Aviv
E. Hochman
Affiliation:
University of Tel-Aviv
*
*Now at Northwestern University, Evanston, Illinois.
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Abstract

Jensen gave a lower bound to Eρ(T), where ρ is a convex function of the random vector T. Madansky has obtained an upper bound via the theory of moment spaces of multivariate distributions. In particular, Madansky's upper bound is given explicitly when the components of T are independent random variables. For this case, lower and upper bounds are obtained in the paper, which uses additional information on T rather than its mean (mainly its expected absolute deviation about the mean) and hence gets closer to Eρ(T).

The importance of having improved bounds is illustrated through a nonlinear programming problem with stochastic objective function, known as the “wait and see” problem.

Keywords

BOUNDSCONVEX FUNCTIONSMEANABSOLUTE DEVIATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 9 , Issue 4 , December 1972 , pp. 803 - 812
Copyright
Copyright © Applied Probability Trust 1972 

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Footnotes

Research done at the Center of Research for Agricultural Economics, Rehovot.

References

[1] Avriel, M. and Williams, A. C. (1970) The value of information and stochastic programming. Operations Res. 18, 947–954.CrossRefGoogle Scholar
[2] Jensen, J. L. W. V. (1906) Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta Math. 30, 175–193.CrossRefGoogle Scholar
[3] Madansky, A. (1959) Bounds on the expectations of a convex function of a multivariate random variable. Ann. Math. Statist. 30, 743–746.Google Scholar
[4] Mangasarian, O. L. (1964) Nonlinear programming problems with stochastic objective functions. Management Sci. 12, 353–359.Google Scholar