  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.
