Inequalities for probability contents of convex sets via geometric average

Moshe Shaked and Y. L. Tong

Journal of Multivariate Analysis, 1988, vol. 24, issue 2, 330-340

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.

Keywords: probability; inequalities; majorization; probability; content; of; rectangles; and; ellipsoids; Schur-concavity; log-concavity; functions; decresing; in; transposition; multivariate; unimodality; elliptically; contoured; distributions; peakedness (search for similar items in EconPapers)
Date: 1988
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