Splitting the Sample at the Largest Uncensored Observation

TL;DR

This paper derives finite sample and asymptotic distributions for the largest censored and uncensored survival times, providing insights into censored data structure and methods for testing follow-up sufficiency.

Contribution

It introduces a structural result showing the sample splits into independent parts based on the largest uncensored time, aiding in survival data analysis.

Findings

Explicit finite sample formulas for survival statistics

Asymptotic distributions derived using extreme value theory

Structural insight into censored survival data

Abstract

We calculate finite sample and asymptotic distributions for the largest censored and uncensored survival times, and some related statistics, from a sample of survival data generated according to an iid censoring model. These statistics are important for assessing whether there is sufficient followup in the sample to be confident of the presence of immune or cured individuals in the population. A key structural result obtained is that, conditional on the value of the largest uncensored survival time, and knowing the number of censored observations exceeding this time, the sample partitions into two independent subsamples, each subsample having the distribution of an iid sample of censored survival times, of reduced size, from truncated random variables. This result provides valuable insight into the construction of censored survival data, and facilitates the calculation of explicit finite sample formulae. We illustrate for distributions of statistics useful for testing for sufficient followup in a sample, and apply extreme value methods to derive asymptotic distributions for some of those.