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Inequalities for probability contents of convex sets via geometric average

Author

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  • Shaked, Moshe
  • Tong, Y. L.

Abstract

It is shown that: If (X1, X2) is a permutation invariant central convex unimodal random vector and if A is a symmetric (about 0) permutation invariant convex set then P{(aX1, X2/a) [set membership, variant] A} is nondecreasing as a varies from )+ to 1 and is non-increasing as a varies from 1 to [infinity] (that is, P{(a1X1, a2X2) [epsilon] A} is a Schur-concave function of (log a1, log a2). Some extensions of this result for the n-dimensional case are discussed. Applications are given for elliptically contoured distributions and scale parameter families.

Suggested Citation

  • Shaked, Moshe & Tong, Y. L., 1988. "Inequalities for probability contents of convex sets via geometric average," Journal of Multivariate Analysis, Elsevier, vol. 24(2), pages 330-340, February.
    • Handle: RePEc:eee:jmvana:v:24:y:1988:i:2:p:330-340
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