## 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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