Strong consistency properties of nonparametric estimators for randomly censored data,I: The product-limit estimator

In reliability and survival-time studies one frequently encounters the followingrandom censorship model:X ₁,Y ₁,X ₂,Y ₂, is an independent sequence of nonnegative rv's, theX _n' s having common distributionF and theY _n' s having common distributionG, Z _n =min{X _n ,Y _n },T _n =IX _n <-Y _n ]; ifX _n represents the (potential) time to death of then-th individual in the sample andY _n is his (potential) censoring time thenZ _n represents the actual observation time andT _n represents the type of observation (T _n =O is a censoring,T _n =1 is a death). One way to estimateF from the observationsZ ₁.T ₁,Z ₂,T ₂, (and without recourse to theX _n' s) is by means of theproduct limit estimator (Kaplan andMeier 6]). It is shown that a.s., uniformly on 0,T] ifH(T ^–)<1 wherel–H=(l–F) (l–G), uniformly onR if whereT _F =sup {x:F(x)<1}; rates of convergence are also established. These results are used in Part II of this study to establish strong consistency of some density and failure rate estimators based on .The third author's research was partly supported by National Research Council of Canada