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Probability Inequalities for n-Dimensional Rectangles via Multivariate Majorization

  • Y. L. Tong

Abstract

Inequalities for the probability content \( P\left[ { \cap _{j\, = \,1}^n\left\{ {\,{a_{1j\,}}\, \leqslant \,{X_j}\, \leqslant \,{a_{2j}}} \right\}} \right] \) are obtained, via concepts of multivariate majorization (which involves the diversity of elements of the 2xn matrix A = (a ij)). A special case of the general result is that \( P\left[ { \cap _{j = 1}^n\left\{ {{a_{1j}} \leqslant {X_j} \leqslant {a_{2j}}} \right\}} \right] \leqslant P\left[ { \cap _{j = 1}^n\left\{ {{{\bar a}_1} \leqslant {X_j} \leqslant {{\bar a}_2}} \right\}} \right] \) for \( {\bar a_{i\,}} = \,\frac{1}{n}\,\sum\nolimits_{j = 1}^n {{a_{ij}}\,\left( {i\, = \,1,\,2} \right).} \). The main theorems apply in most important cases, including the exchangeable normal, t, chi-square and gamma, F, beta, and Dirichlet distributions. The proofs of the inequalities involve a convex combination of an n-dimensional rectangle and its permutation sets.

Keywords

Convex Combination Stochastic Matrix Probability Inequality Weak Concept Chain Majorization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York, Inc. 1989

Authors and Affiliations

  • Y. L. Tong
    • 1
  1. 1.Georgia Institute of TechnologyUSA

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