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Bivariate Extension of the Method of Polynomials for Bonferroni-Type Inequalities

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  • Galambos, J.
  • Xu, Y.

Abstract

Let A1, A2, ..., An and B1, B2, ..., BN be two sequences of events. Let mn(A) and mN(B) be the number of those Aj and Bk, respectively, which occur. Set Sk,t for the joint (k, t)th binomial moment of the vector (mn(A),mN(B)). We prove that linear bounds in terms of the Sk,t on the distribution of the vector (mn(A),mN(B)) are universally true if and only if they are valid in a two dimensional triangular array of independent events Aj and Bi with P(Aj) = p and P(Bi) = s for all j and i. This allows us to establish bounds on P(mn(A) = u, mN(B) = v) from bounds on P(mn - u(A) = 0, mN - v(B) = 0). Several new inequalities are obtained by using our method.

Suggested Citation

  • Galambos, J. & Xu, Y., 1995. "Bivariate Extension of the Method of Polynomials for Bonferroni-Type Inequalities," Journal of Multivariate Analysis, Elsevier, vol. 52(1), pages 131-139, January.
    • Handle: RePEc:eee:jmvana:v:52:y:1995:i:1:p:131-139
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    Cited by:

    1. Barakat, H.M., 2009. "Multivariate order statistics based on dependent and nonidentically distributed random variables," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 81-90, January.
    2. Simonelli, Italo, 1999. "An Extension of the Bivariate Method of Polynomials and a Reduction Formula for Bonferroni-Type Inequalities," Journal of Multivariate Analysis, Elsevier, vol. 69(1), pages 1-9, April.
    3. Qin Ding & Eugene Seneta, 2017. "Bivariate Binomial Moments and Bonferroni-Type Inequalities," Methodology and Computing in Applied Probability, Springer, vol. 19(1), pages 331-348, March.

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