Limit theorems for the ratio of the Kaplan-Meier estimator or the Altshuler estimator to the true survival function

摘要: LetX1,X2, ...,Xn be a sequence of nonnegative independent random variables with common continuous distribution functionF. LetY1,Y2, ...,Yn another functionG, also {Xi}. We can only observeZi=min(Xi,Yi), and\(\delta _i = I_{(X_i \leqslant Y_i )} \). LetH=1−(1−F)(1−G) the function ofZ. In this paper, limit theorems for ratio Kaplan-Meier estimator\(\hat S_n (t)\) or Altshuler estimator\(\tilde to true survival functionS(t) are given. It is shown that (1)P(δ(n)=1 i.o.)=0 ifF(τH) 1 where δ(n) corresponding indicator of\(T Z_{(n)} \mathop {\max }\limits_{1 i n} Z_i ,\tau _H \inf \{ t:H(t) 1\} ;(2)\mathop {\sup }\limits_{t T_n } \left| {\frac{{\hat (t)}}{{S(t)}} - 1} \right|,\mathop {\frac{{S(t)}}{{\hat (t)}} {\frac{{\tilde \right|\) and\(\mathop have same order\(O\left( {n^{\frac{{ + \alpha }}{2}} (\log n)^{\frac{{1 \beta \log n)^{ \frac{\gamma }{2}} \right)\) a.s., {Tn} constants such 1−H(Tn)=n−α(logn)β(log logn)γ.