On Aggregation of Uncensored and Censored Observations

Abstract

In survival analysis a random right-censoring partitions data into uncensored and censored observations of the lifetime of interest. The dominance of uncensored observations is a familiar methodology in nonparametric estimation motivated by the classical Kaplan–Meier product-limit and Cox partial likelihood estimators. Nonetheless, for high rate censoring it is of interest to understand what, if anything, can be done by aggregating uncensored and censored observations for the staple nonparametric problems of density and regression estimation. The oracle, who knows distribution of the censoring lifetime, can use each subsample for consistent estimation and hence may shed light on the aggregation. The oracle’s asymptotic theory reveals that density estimation, based on censored observations, is an ill-posed problem with slower rates of risk convergence, the ill-posedness occurs in frequency-domain, its severity increases with frequency, and accordingly a special aggregation on low frequencies may be beneficial. On the other hand, censored observations are not ill-posed for nonparametric regression and the aggregation is feasible. Based on these theoretical results, methodology of aggregation in frequency domain is developed and proposed estimators are tested on simulated and real examples.