On the Koziol—Green model for random censorship

Abstract

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, Csörgó and Horv´th, 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 functional including the Cramér-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.