On the effect of the Kaplan-Meier estimator’s assumed tail behavior on goodness-of-fit testing

Abstract

When analyzing lifetime data in the presence of censoring one is often required to estimate the distribution function of the lifetimes non-parametrically. The most popular estimator used for this purpose is the Kaplan-Meier estimator. Interestingly, in its initial formulation this estimator is only defined up to the observed sample maximum. For values larger than the sample maximum two different assumptions are commonly used in the statistical literature. The first is to set the value of the estimate to one while the second is to use the value of the estimate at the sample maximum when estimating the tail of the distribution function. This paper illustrates the profound effect of these assumptions on the sizes and powers of goodness-of-fit tests for three classes of distributions often used in survival analysis. These differences are illustrated using observed remission time data. The considered classes of distributions are the exponential, Weibull and gamma. As a result of independent interest, we amend two classes of tests developed for the gamma distribution in the full sample case for use with censored data.