The existence of immune or cured individuals in a population and whether there is sufficient follow-up in a sample of censored observations on their lifetimes to be confident of their presence are questions of major importance in medical survival analysis. So far only a few candidates have been put forward as possible test statistics for the existence of sufficient follow-up in a sample. Here we discuss properties of one such statistic, $Q_n$. Assuming an iid censoring model, we obtain a formula for the finite sample distribution of $Q_n$ which we use to find its asymptotic distribution under scenarios of sufficient or insufficient followup.
A new and very useful finding is that the asymptotic distribution is parameter free in the null case when follow-up is insufficient. We illustrate with application to a glioma cancer data set.