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Product-Type Probability Bounds of Higher Order

Published online by Cambridge University Press:  27 July 2009

Hentry W. Block ,
Tim Costigan  and
Allan R. Sampson
Hentry W. Block
Affiliation:
Department of Mathematics and StaatisticsUniversity of Pittsburgh Pittsburgh, Pennsylvania 15260
Tim Costigan
Affiliation:
Department of StatisticsThe Ohio State University, 141 Cockins Hall 1958 Neil Avenue Columbus, Ohio 43210-1247
Allan R. Sampson
Affiliation:
Department of Mathematics and StatisticsUniversity of Pittsburgh Pittsburgh, Pennsylvania 15260
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Abstract

GIaz and Johnson [14] introduce ith-order product-type approximations, βi, i =1,…n − 1, for Pn = P(X1 ≤c1, X2 ≤ c2,…Xn≤ cn) and show that Pn≥ βn−1 ≥ βn−2 ≥… ≥ β2 ≥ β1 when X is MTP2. In this article, it is shown that

under weaker positive dependence conditions. For multivariate normal distributions, these conditions reduce to cov(Xi,Xj) ≥ 0 for 1 ≤ i < j ≤ n and cov(Xi,Xj| Xj−1) ≥ 0 for 1 ≤ i < j − 1, j = 3,…,n. This is applied to group sequential analysis with bivariate normal responses. Conditions for Pn ≥ β3 ≥ β2 ≥β1 are also derived. Bound conditions are also obtained that ensure that product-type approximations are nested lower bounds to upper orthant probabilities P(X1 > C1,…,Xn > cn). It is shown that these conditions are satisfied for the multivariate exponential distribution of Marshall and Olkin [20].

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 6 , Issue 3 , July 1992 , pp. 349 - 370
Copyright
Copyright © Cambridge University Press 1992

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