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

This paper gives sharp bounds for the joint survival function G(t(1), t(2),...,t(r)) identical with P(X(1) > t(1), X(2) > t(2),...,X(r) > t(r)), and for the marginal survival functions S(j)(t) identical with P(X(j) > t), j = 1,2,...,r, when the sub-survival functions S(j) (*)(t) identical with P(X(j) > t, X(j) = min(k=1,2),...,(r)X(k)) 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.