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In this paper we consider a time-to-event variable T that is subject to random right censoring, and we assume that the censoring time C is stochastically dependent on T and that there is a positive probability of not observing the event. There are various situations in practice where this happens, and appropriate models and methods need to be considered to avoid biased estimators of the survival function or incorrect conclusions in clinical trials. We consider a fully parametric model for the bivariate distribution of (T, C), that takes these features into account. The model depends on a parametric copula (with unknown association parameter) and on parametric marginal distributions for T and C. Sufficient conditions are developed under which the model is identified, and an estimation procedure is proposed. In particular, our model allows to identify and estimate the association between T and C, even though only the smallest of these variables is observable. The finite sample performance of the estimated parameters is illustrated by means of a thorough simulation study and the analysis of breast cancer data.
Delhelle et al. (Tue,) studied this question.