On the Koziol-Green model for random censorship

SUMMARY In the Koziol-Green (1976) model of random censorship the survival distribution of the censoring variables is some power, the censoring parameter, of the survival distribution of the lifetimes. Using a strong approximation result of Burke, Csorgo and Horvath, we construct empirical exact confidence bands for the life distribution of the censored variable, and a randomly Efron-transformed variant of the usual product-limit process is shown to converge weakly to the Brownian motion process on any interval [0, T] when the censoring parameter is also estimated from the sample. The technique automatically gives rates of convergence for a wide class of functionals including the Cramer-von Mises functional whose limit theory is worked out in detail. The latter goodness-of-fit statistic is applied to reexamine the hospital data of Koziol & Green (1976) to test whether oestrogen treatment for prostatic cancer was effective or not for the cancer itself while it caused censoring deaths of cardiovascular diseases.