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A note on the uniform consistency of the Kaplan-Meier estimator. (English) Zbl 0631.62043

Let \(\{X_ n\), \(n\geq 1\}\) be i.i.d. with \(P(X_ i\leq u)=F(u)\) and \(\{U_ n\), \(n\geq 1\}\) be i.i.d. with \(P(U_ i\leq u)=G(u)\). \(\hat F_ n(t)\) is the Kaplan-Meier estimator based on the censored data \((\tilde X_ i=X_ i\wedge U_ i\), \(\delta_ i=1_{(X_ i\leq U_ i)}\), \(1\leq i\leq n)\). In this note, it is shown that for \(T_ n=\max_{1\leq i\leq n}\tilde X_ i\), \[ -\lim_{n\to \infty}\sup_{t\leq T_ n}| \hat F_ n(t)-F(t)| =0. \] Hence, the largest interval on which the Kaplan-Meier estimator is uniformly consistent is found.

MSC:

62G05 Nonparametric estimation
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