Constrained Nonparametric Estimation of Survival Functions under Censorings and Truncations

Abstract

In survival analysis, possible complications include ordering constraints and missing data due to grouping, censoring, or truncation. Dykstra [JASA 77 (1982)] and Dykstra, Kochar, and Robertson [Ann. Stat. 19 (1991)] derived the closed-form nonparametric MLE's of survival functions with right-censored observations under stochastic and uniformly stochastic ordering constraints. However, it cannot be extended to other type of censored data. Turnbull [J. Roy. Statist. Soc. Ser.B 38, (1976)] suggested an ingenious, iterative procedure for computing the nonparametric MLE's of survival functions under arbitrary grouping, censoring and truncation which has been incorporated into statistical software such as Minitab. We proposed a two-step-constrained procedure by generalizing Turnbull's procedure to the case where general, distributional constraints can be added. Though the estimating procedure is non-parametric in nature, the constraint step is incorporated through multinomial constraint procedures which are well understood and are quite tractable. It is shown that this two-step-constrained procedure does converge to the correct restricted MLE's under linear, stochastic, uniformly stochastic, and likelihood ratio ordering constraints. With standard regularity conditions, it will work on any convex constraint region.