Strong Consistency of a Nonparametric Estimator of the Survival Function with Doubly Censored Data

Abstract

A double censoring mechanism is such that each variable $X$ in the sample is observable if and only if $X$ is within the observation interval $\lbrack Z, Y \rbrack$. Otherwise, we can only determine whether $X$ is less than $Z$ or greater than $Y$ and observe $Z$ or $Y$ correspondingly. This kind of censoring occurs often in collecting lifetime data. Our problem is to estimate the survival function of $X, S_X(t) = P \lbrack X > t \rbrack$, from a doubly censored sample, where $X$ is assumed to be independent of the random interval $\lbrack Z, Y \rbrack$. We establish sufficient conditions for which $S_X(t)$ is identifiable and then prove the strong consistency of the self-consistent estimator $\hat{S}_X(t)$ for $S_X(t)$. This investigation generalizes the results available for the right censored data.