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Sharp Bounds on the Largest of some Linear Combinations of Random Variables with Given Marginal Distributions

Published online by Cambridge University Press:  27 July 2009

Isaac Meilijson
Isaac Meilijson
Affiliation:
School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel
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Abstract

Let X be a random vector and A a matrix. Let M be the maximal coordinate of the vector AX. For given marginal distributions of the coordinates of X, we present sharp bounds on the expectations of convex increasing functions of M. We derive joint distributions of X that achieve some of these bounds, and under these “worst case” distributions we study the joint distribution of M and the index of the largest coordinate of AX. Some possible applications are PERT network analysis and design of experiments.

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 5 , Issue 1 , January 1991 , pp. 1 - 14
Copyright
Copyright © Cambridge University Press 1991

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References

Bellman, R. (1960). Introduciton to matrix analysis. New York: McGraw-Hill.Google Scholar
Klein, Haneveld W. K. (1985). Robustness against dependence in PERT: An application of duality and distributions with given marginals. Mathematical Programming Study. North Holland.Google Scholar
Lasdon, L.S. (1970). Optimization theory for large systems. Macmillan series in operation research. London: Collier-MacMillan.Google Scholar
Marshall, A. W. & Olkin, I. (1961). Game theoretic proof that Chebyshev inequalities are sharp. Pacific Journal of Mathematics 11 (4): 1421–1429.CrossRefGoogle Scholar
Meilijson, I. & Nádas, A. (1979). Convex majorization with an application to the length of critical paths. Journal of Applied Probability 16: 671–677.CrossRefGoogle Scholar
Nádas, A. (1979). Probabilistic PERT. IBM Journal of Research and Development 23: 339–347.Google Scholar
Ross, S. M. (1983). Stochastic processes. New York: John Wiley.Google Scholar
Rüschendorf, L. (1981). Sharpness of Fréchet-bounds. Zeitschrift Wahr. Verw. Geb. 57, 293–302.Google Scholar
Tiit, E.-M. (1984). Construction of distributions with given marginals and correlation matrix. Acta Comm. University of Tartu (Estonia) 685: 21–36.Google Scholar
Weiss, G. (1986). Stochastic bounds on distributions of optimal value functions with applications to PERT, network flow and reliability. Operations Research 34 (4): 595–605.Google Scholar