Product-Type Probability Bounds of Higher Order

Abstract

GIaz and Johnson [14] introduce ith-order product-type approximations, β i , i =1,… n − 1, for P n = P ( X 1 ≤ c 1 , X 2 ≤ c 2 ,… X n ≤ c n ) and show that P n ≥ β n −1 ≥ β n −2 ≥… ≥ β 2 ≥ β 1 when X is MTP 2 . In this article, it is shown that under weaker positive dependence conditions. For multivariate normal distributions, these conditions reduce to cov( X i , X j ) ≥ 0 for 1 ≤ i < j ≤ n and cov( X i , X j | X j−1 ) ≥ 0 for 1 ≤ i < j − 1 , j = 3,…, n . This is applied to group sequential analysis with bivariate normal responses. Conditions for P n ≥ β 3 ≥ β 2 ≥β 1 are also derived. Bound conditions are also obtained that ensure that product-type approximations are nested lower bounds to upper orthant probabilities P ( X 1 > C 1 ,…, X n > c n ). It is shown that these conditions are satisfied for the multivariate exponential distribution of Marshall and Olkin [20].