期刊名称：Proceedings of the National Academy of Sciences
印刷版ISSN：0027-8424
电子版ISSN：1091-6490
出版年度：1976
卷号：73
期号：1
页码：11-13
DOI：10.1073/pnas.73.1.11
语种：English
出版社：The National Academy of Sciences of the United States of America
摘要：This paper gives sharp bounds for the joint survival function G(t1, t2...,tr) {equiv} P(X1 > t1, X2 > t2...,Xr > tr), and for the marginal survival functions Sj(t) {equiv} P(Xj > t), j = 1,2...,r, when the sub-survival functions Sj*(t) {equiv} P(Xj > t, Xj = mink=1,2...,rXk) are fixed. Theorem 1 gives the bounds for r = 2, and Theorem 2 gives the bounds for general r. Theorem 3 applies the result to the competing risks problem, and presents empirical bounds based on the observations. Finally, an example illustrates the bounds.
