Extremes of censored and uncensored lifetimes in survival data

We consider a random censoring model for survival analysis, allowing the possibility that only a proportion of individuals in the population are susceptible to death or failure, and the remainder are immune or cured. Susceptibles suffer the event under study eventually, but the time at which this occurs may not be observed due to censoring. Immune individuals have infinite lifetimes which are always censored in the sample. Assuming that the distribution of the susceptibles’ lifetimes as well as the censoring distribution have infinite right endpoints and are in the domain of attraction of the Gumbel distribution, we obtain asymptotic distributions, as sample size tends to infinity, of statistics relevant to testing for the possible existence of immunes in the population.