We extend Cox's classical regression model to accomodate partially informative censored data. In this type of data, each observation is the minimum of one lifetime and two censoring times. The survival function of one of these censoring times is a power of the survival function of the lifetime. We call this the informative censoring time. The distribution of the other censoring time has no relation with the distribution of the lifetime. It is called the non-informative censoring time. In this

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... del we specify a semiparametric relation between the lifetime and a covariate where we take into account that also informatively censored observations contribute to this relation. We introduce an estimator for the cumulative baseline hazard function and use maximum likelihood techniques for the estimation of the parameters in the model. Our main results are strong consistency and asymptotic normality of these estimators. The proof uses the general theory of Murphy and van der Vaart (2000) on profile likelihoods. Finally the method is applied to simulated data and to real data examples on survival with malignant melanoma and survival after bone marrow transplantation.