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A Nonsymmetric Correlation Inequality for Gaussian Measure

Author

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  • Szarek, Stanislaw J.
  • Werner, Elisabeth

Abstract

Let[mu]be a Gaussian measure (say, onRn) and letK,L[subset, double equals]Rnbe such thatKis convex,Lis a "layer" (i.e.,L={x: a[less-than-or-equals, slant] [less-than-or-equals, slant]b} for somea, b[set membership, variant]Randu[set membership, variant]Rn), and the centers of mass (with respect to[mu]) ofKandLcoincide. Then[mu](K[intersection]L)[greater-or-equal, slanted][mu](K)·[mu](L). This is motivated by the well-known "positive correlation conjecture" for symmetric sets and a related inequality of Sidak concerning confidence regions for means of multivariate normal distributions. The proof uses the estimate[Phi](x)> 1-((8/[pi])1/2/(3x+(x2+8)1/2))e-x2/2,x>-1, for the (standard) Gaussian cumulative distribution function, which is sharper than the classical inequality of Komatsu.

Suggested Citation

  • Szarek, Stanislaw J. & Werner, Elisabeth, 1999. "A Nonsymmetric Correlation Inequality for Gaussian Measure," Journal of Multivariate Analysis, Elsevier, vol. 68(2), pages 193-211, February.
    • Handle: RePEc:eee:jmvana:v:68:y:1999:i:2:p:193-211
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    References listed on IDEAS

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    1. Koldobsky, A. L. & Montgomery-Smith, S. J., 1996. "Inequalities of correlation type for symmetric stable random vectors," Statistics & Probability Letters, Elsevier, vol. 28(1), pages 91-97, June.
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    Cited by:

    1. Enkelejd Hashorva & Jürg Hüsler, 2003. "On multivariate Gaussian tails," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 55(3), pages 507-522, September.
    2. Kameng Nip & Zhenbo Wang & Zizhuo Wang, 2021. "Assortment Optimization under a Single Transition Choice Model," Production and Operations Management, Production and Operations Management Society, vol. 30(7), pages 2122-2142, July.
    3. Hashorva, Enkelejd, 2002. "Remarks on domination of maxima," Statistics & Probability Letters, Elsevier, vol. 60(1), pages 101-109, November.
    4. Shao, Qi-Man, 2003. "A Gaussian correlation inequality and its applications to the existence of small ball constant," Stochastic Processes and their Applications, Elsevier, vol. 107(2), pages 269-287, October.

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    1. Shao, Qi-Man, 2003. "A Gaussian correlation inequality and its applications to the existence of small ball constant," Stochastic Processes and their Applications, Elsevier, vol. 107(2), pages 269-287, October.

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