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By approximating the classical product-limit estimator of a distribution function with an average of iid random variables, we derive sufficient and necessary conditions for the rate of (both strong and weak) uniform convergence of the product-limit estimator over the whole line. These findings somehow fill a longstanding gap in the asymptotic theory of survival analysis. The result suggests a natural way of estimating the rate of convergence. We also prove a related…

删失数据下自助法的近似精度AI 解读

In survival analysis with censored data, we consider three closely related survival function estimators: the Kaplan-Meier, Nelson and moment estimators. We derive the Edgeworth expansions for these three estimators with Studentization. Edgeworth expansions for the corresponding bootstrap statistics are also given. It is found that the bootstrap approximation is better than the normal approximation for the Studentized Kaplan-Meier and Nelson estimators, but not so for the…

截断数据 Lynden–Bell 估计量的强一致收敛AI 解读

In this paper, we prove that the Lynden-Bell estimator of a distribution function in the random truncation model is uniformly strong consistent over the whole half line, a problem left open by Woodroofe.

通过新解释从物种问题推广到一般覆盖问题AI 解读

A basic interpretation is given which provides a new way of understanding the structure of the species problem and which leads to the popular Turing-Good-Robbins estimator. Through this interpretation we provide an explanation why the Turing-Good-Robbins estimators are always biased. An iterative procedure is suggested and applied to these estimators, which leads to new estimators whose biases are reduced. Using this basic construction we are able to generalize our…

右删失两样本处理效应一般估计量的大样本性质AI 解读

The estimation of the treatment effect in the two-sample problem with right censoring is of interest in survival analysis. In this article we consider both the location shift model and the scale change model. We establish the large-sample properties of a generalized Hodges-Lehmann type estimator. The strong consistency is established under the minimal possible conditions. The asymptotic normality is also obtained without imposing any conditions on the censoring mechanisms.…

严重删失数据下处理效应的一般估计量AI 解读

A generalized Hodges-Lehmann type estimator for the treatment effect in the two-sample problem with right censoring, is proposed based on an inverse-quantile-type idea using truncated versions of the Kaplan-Meier estimators over the subspace where they are consistent. Its strong consistency and asymptotic normality can be obtained, under no conditions on the uninformative censorings, and the resulting variance is easily estimable from the data. In simulation studies the…

重度删失数据下处理效应的一般估计量AI 解读

A generalized Hodges-Lehmann type estimator for the treatment effect in the two-sample problem with right censoring, is proposed based on an inverse-quantile-type idea using truncated versions of the Kaplan-Meier estimators over the subspace where they are consistent. Its strong consistency and asymptotic normality can be obtained, under no conditions on the uninformative censorings, and the resulting variance is easily estimable from the data. In simulation studies the…

截断数据非参数最大似然估计量的若干表示AI 解读

The nonparametric maximum likelihood estimators of the distribution functions of observations in the truncation model are represented as iid mean processes, with a remainder term of order $o(n^{-1/2})$ a.s.

单峰分布函数的估计AI 解读

This paper deals with the problem of efficiently estimating (asymptotically minimax) a distribution function when essentially nothing is known about it except that it is unimodal. The sample distribution function $F_n$ is shown to be asymptotically minimax among the family $\mathscr{E}$ of all unimodal distribution functions. Since $F_n$ does not belong to this family, estimators belonging to this family are constructed and are shown to be asymptotically minimax relative to…

对称分布的分位数估计AI 解读

The problem is to estimate a quantile of a symmetric distribution. The cases of known and unknown center are studied for small and large samples. The estimators for known center are the sample quantile, the symmetrized sample quantile, the sample quantile from flipped over data, the Rao-Blackwellized sample quantile, and a Bayes estimator using a Dirichlet prior. For center unknown, we study the analogues of the first four estimators listed above. For small samples and…

对称分布的估计AI 解读

Suppose that $F_0$ is a population which is symmetric about zero, so that $F(\cdot) = F_0(\cdot - \theta)$ is symmetric about $\theta$. We consider the problem of estimating $F_0$ (shape parameter), both $\theta$ and $F_0$, and $F$ based on a random sample from $F$. First, some asymptotically minimax bounds are obtained. Then, some estimates are constructed which are asymptotically minimax-efficient (the risks of which achieve the minimax bounds uniformly). Furthermore, it…