二维Kaplan-Meier估计a.s.一致收敛速度

The Strong Consistent Convergence Rate of Kaplan-Meier Estimation in Two Dimensions

设(T_(11),T_(21)),…,(T_(1n),T_(2n))和(Y_(11),Y_(21)),…,(Y_(1n),Y_(2n))都是i.i.d.的非负r.v.F(s,t)=P(T_(11)>s,T_(21)>t)和G(s,t)=P(Y_(11)>s,T_(21)>t)都是未知的连续函数.文[3]给出了基于观察Z_(if),=min(T_(if),T_(if))和δ_(if)=I(T_(if)≤Y_(if)),i=1,2,i=1,…,n估计F的二维Kaplan-Meier估计_n.本文在一定条件下证明了_n a.s.一致收敛于F的速度为。

Let (T11,T21)), …, (T1n, T2n)bci.i.d.P(T1tt >s, T2t>t) - F(s, t) and (y11, K21), …, (y1n, Y2n) be i.i.d.P(Y 11 > s ,Y2t>t) = G(s, t), where both F and G are unknown continuous. For i = 1 , 2 , j = 1, -, n set zij = min(T1f, Y1f and δif = I(Tij ≤ Yij). One way to estimate F from the observations {(z1l, z2l, δ1l,δ2l)}f=1n, is so-called Kaplan-Meier estimator, Fn[3]. In this paper it is shown that Fn is uniformly a..v. Consisistent with rate O (logn / n ). that isp under some conditions.