A Note on the Uniform Consistency of the Kaplan-Meier Estimator

Let ₙ, n ≥ 1\ be i.i.d with P(Xᵢ ≤ u) = F(u) and ₙ, n ≥ 1\ be i.i.d. with P(Uᵢ ≤ u) = G(u). F̂ₙ(t) is the Kaplan-Meier estimator based on the censored data (X̃ᵢ = Xᵢ Uᵢ, δᵢ = 1(Xᵢ ≤ Uᵢ), 1 ≤ i ≤ n). In this note, it is shown that for Tₙ = max1 ≤ i ≤ n X̃ᵢ, pr-limn → ∞ t ≤ Tₙ |F̂ₙ(t) - F(t)| = 0. Hence, the largest interval on which the Kaplan-Meier estimator is uniformly consistent is found.