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This point is an immediate consequence of the likelihood principle. See Berger and Wolpert (1988, Ch. 4). 326), Greenhouse (1982), and Hill (1987a). Based upon the above discussion, the desired probabilities may be written as Pr(Z E Ii I z) (2) for i = 0, ... , n. Although evaluation of (2), which is conditional upon the exact times both of death and of censoring, is our goal, we will in fact calculate an approximation, and then use upper and lower bounds for the probabilities in (2). Instead of conditioning on the exact censoring times (1), we will condition only on the intervals 10 in which censoring occurred.

These theorems make rather transparent the relationship between the KM and BH evaluations. By noting the relationship between A(i) and AKM(i) as given in Theorems 1 and 2, it may be observed that if we evaluate the BH estimator at the XCi), but with an :LJ=o 34 adjusted at risk vector that subtracts 1 from each element of the original at risk vector n, then we obtain the KM estimator; and conversely, if we add 1 to the original at risk vector, and evaluate the KM estimator for this modified at risk vector, then we obtain the BH estimator.

Let AKM(i) = [N _ (i _ 1)1 - C(i - 1)] , for i = 1, ... , n, and set AKM(O) by the KM estimator are = O. Then the estimated probabilities for the Ii as given and for i = 0, ... , n - 1, P(i + 1) = [1 - AKM(O)] x ... x [l - AKM(i)] x AKM(i + 1) . The tail masses beyond the observed death times are SKM(i) = i II [l - AKM(S)], for i = 0, ... , n .. s=O Proof The number at risk at the time of the ith death is ni = N - (i - 1) - C(i - 1). The number of deaths at X(i) is di = 1. Hence by the usual formula for the KM estimator, the survival function at t such that XU) < t ::;; xU+ 1) is 33 II (1 - ~!