Limit theorems for the ratio of the Kaplan-Meier estimator or the Altshuler estimator to the true survival function期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:	Zheng Zukang

Institution:	(1) Department of Statistics and Operations Research, Fudan University, 200433 Shanghai, China

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

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