Zheng Zukang
Let
X1,
X2,...,
Xn be a sequence of nonnegative independent random variables with a common continuous distribution function
F.Let
Y1,
Y2,...,
Yn be another sequence of nonnegative independent random variables with a common continuous distribution function
G,also independent of {
Xi}.We can only observe
Zi=min(
Xi,
Yi),and
} $)
.Let
H=1-(1-F)(1-G) be the distribution function of
Z.In this paper,the limit theorems for the ratio of the Kaplan-Meier estimator
Ŝn(
t) or the Altshuler estimator
Sn(
t) to the true survival function
S(t) are given.It is shown that (1)
P(δ
(n)=1 i.o.)=0 if
F(τ
H)<1 and
P(δ
n=0 i.o.)=0 if
G(τ
H) > 1 where δ
(n) is the corresponding indicator function of
} = \mathop {\max }\limits_{1 \leqslant i \leqslant n} Z_i,\tau _H = \inf \{ t:H(t) = 1\} ;(2)\mathop {\sup }\limits_{t \leqslant T_n } \left| {\frac{{\hat S_n (t)}}{{S(t)}} - 1} \right|,\mathop {\sup }\limits_{t \leqslant T_n } \left| {\frac{{S(t)}}{{\hat S_n (t)}} - 1} \right|,\mathop {\sup }\limits_{t \leqslant T_n } \left| {\frac{{\tilde S_n (t)}}{{S(t)}} - 1} \right|$)
and
}}{{\hat S_n (t)}} - 1} \right|$)
have the same order
^{\frac{{1 - \beta }}{2}} (\log \log n)^{ - \frac{\gamma }{2}} } \right)$)
a.s.,where {
Tn} is a sequence of constants such that 1-
H(Tn)=
n-α(log
n)
β(log log
n)
γ.
