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Proceedings of the National Academy of Sciences of the United States of America logoLink to Proceedings of the National Academy of Sciences of the United States of America
. 1976 Jan;73(1):11–13. doi: 10.1073/pnas.73.1.11

Bounds for a joint distribution function with fixed sub-distribution functions: Application to competing risks

Arthur V Peterson Jr 1,*
PMCID: PMC335828  PMID: 16592299

Abstract

This paper gives sharp bounds for the joint survival function G(t1, t2,...,tr) ≡ P(X1 > t1, X2 > t2,...,Xr > tr), and for the marginal survival functions Sj(t) ≡ P(Xj > t), j = 1,2,...,r, when the sub-survival functions Sj*(t) ≡ 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.

Keywords: survival functions, censored data

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Selected References

These references are in PubMed. This may not be the complete list of references from this article.

  1. Tsiatis A. A nonidentifiability aspect of the problem of competing risks. Proc Natl Acad Sci U S A. 1975 Jan;72(1):20–22. doi: 10.1073/pnas.72.1.20. [DOI] [PMC free article] [PubMed] [Google Scholar]

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