On nonparametric likelihood ratio estimation of survival probabilities for censored data

Abstract

Thomas and Grunkemeier (1975) proposed a nonparametric likelihood ratio method for interval estimation of survival probabilities for randomly censored data. Their method always produces confidence intervals inside [0,1] and has a better performance than the normal approximation method based on the Kaplan-Meier estimate and Greenwood's formula. In this note we show that the likelihood ratio used by Thomas and Grunkemeier (1978) is a “genuine” nonparametric likelihood ratio. That is, it can be derived by considering the parameter space of all survival functions. This property is not shared by many existing empirical likelihood methods. We also note that this result is not a direct consequence of the fact proved by Kaplan and Meier (1958) that the maximum of the likelihood over the space of all survival functions is achieved in the subspace of all discrete survival functions supported on the observed uncensored lifetimes. Another objective of this note is to point out that the likelihood ratio approach can also be used to draw joint inferences on any finite number of probabilities, and test goodness of fit of a given survival function for censored data. A rigorous derivation for the limiting distribution of the likelihood ratio is also provided.