Asymptotic properties of the nonparametric survival curve estimators under variable censoring

Abstract

Let X,₁,Y₁,X₂,Y₂, be an independent sequence of random variables, the X_n,s having common continuous survival function F and the * Y_n’s having continuous survival functions G_n’s /survival function= 1-distribution function/. It is shorn that the product limit estimator of F from data $\{ {Z_{\rm{n}}},{\delta _{\rm{n}}}\} _{{\rm{n}} = 1}^\infty$ where ${Z_{\rm{n}}} = \min ({{\rm{X}}_{\rm{n}}}|{{\rm{Y}}_{\rm{n}}})$ and ${\delta _{\rm{n}}} = [{{\rm{X}}_{\rm{n}}} \le {{\rm{Y}}_{\rm{n}}}]$ / [ 3 denotes the indicator function/, /see Kaplan and Meier, J.A.S.A. 53 /1958/ 457-481/ is strong uniformly consistent on an interval (-∞,T] under some reasonable conditions. Assuming that F is distributed according to a Dirichlet process with parameter ∝ /see Ferguson, Arm. Statist. 1 /1973/ 209-230/ it is shown that the sup distance between the Bayesian estimator /see Susarla, Van Ryzin J.A.S.A. 72 A976/ 889-902/ and the product limit estimator is small enough to remain valid all results using the Bayesian estimator.