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Supermodular Stochastic Orders and Positive Dependence of Random Vectors

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  • Shaked, Moshe
  • Shanthikumar, J. George

Abstract

The supermodular and the symmetric supermodular stochastic orders have been cursorily studied in previous literature. In this paper we study these orders more thoroughly. First we obtain some basic properties of these orders. We then apply these results in order to obtain comparisons of random vectors with common values, but with different levels of multiplicity. Specifically, we show that if the vectors of the levels of multiplicity are ordered in the majorization order, then the associated random vectors are ordered in the symmetric supermodular stochastic order. In the non-symmetric case we obtain bounds (in the supermodular stochastic order sense) on such random vectors. Finally, we apply the results to problems of optimal assembly of reliability systems, of optimal allocation of minimal repair efforts, and of optimal allocation of reliability items.

Suggested Citation

  • Shaked, Moshe & Shanthikumar, J. George, 1997. "Supermodular Stochastic Orders and Positive Dependence of Random Vectors," Journal of Multivariate Analysis, Elsevier, vol. 61(1), pages 86-101, April.
    • Handle: RePEc:eee:jmvana:v:61:y:1997:i:1:p:86-101
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    References listed on IDEAS

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    1. Joe, Harry, 1990. "Multivariate concordance," Journal of Multivariate Analysis, Elsevier, vol. 35(1), pages 12-30, October.
    2. Shaked, M. & Shanthikumar, J. G. & Tong, Y. L., 1995. "Parametric Schur Convexity and Arrangement Monotonicity Properties of Partial Sums," Journal of Multivariate Analysis, Elsevier, vol. 53(2), pages 293-310, May.
    3. Block, Henry W. & Sampson, Allan R., 1988. "Conditionally ordered distributions," Journal of Multivariate Analysis, Elsevier, vol. 27(1), pages 91-104, October.
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