Algebraic Methods Toward Higher-Order Probability Inequalities

  • Kenneth I. Gross
  • Donald St. P. Richards
Part of the Trends in Mathematics book series (TM)


In work motivated by problems in the analysis of variance, Kimball [15]_proved that if Vi = Qi/Q0, i = 1,…,n, where the Q i are mutually independent positive random variables, then V 1,...,V n are positively upper orthant dependent (PUOD),
$$P(\mathop \cap \limits_{i = 1}^n \{ {V_i} \geqslant {v_i}\} ) \geqslant \prod\limits_{i = 1}^n {P({V_i} \geqslant {v_i}),} {v_1}, \ldots ,{v_n} \geqslant 0,$$
and also are positively lower orthant dependent (PLOD),
$$P(\mathop \cap \limits_{i = 1}^n \{ {V_i} \leqslant {v_i}\} ) \geqslant \prod\limits_{i = 1}^n {P({V_i} \leqslant {v_i}),} {v_1}, \ldots ,{v_n} \geqslant 0.$$
These results motivated much research on other inequalities; cf. [4], [10], [20], [24].


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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Kenneth I. Gross
    • 1
  • Donald St. P. Richards
    • 2
  1. 1.Department of Mathematics & StatisticsUniversity of VermontBurlingtonUSA
  2. 2.Division of StatisticsUniversity of VirginiaCharlottesvilleUSA

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