A Strong Representation of the Product-Limit Estimator for Left Truncated and Right Censored Data

In this paper we consider the TJW product-limit estimatorF_n(x) of an unknown distribution functionFwhen the data are subject to random left truncation and right censorship. An almost sure representation of PL-estimatorF_n(x) is derived with an improved error bound under some weaker assumptions. We obtain the strong approximation ofF_n(x)−F(x) by Gaussian processes and the functional law of the iterated logarithm is proved for maximal derivation of the product-limit estimator toF. A sharp rate of convergence theorem concerning the smoothed TJW product-limit estimator is obtained. Asymptotic properties of kernel estimators of density function based on TJW product-limit estimator is given.     

A Berry-Esséen Bound for the Product-Limit Estimator under Left-Truncation and Right-Censoring

For left-truncated and right-censored data, the product-limit estimator F?_xis a well-known nonparametric estimator for the distribution function F_x of the target variable X such as the survival time. Since F?_xas a very complicated product form we establish first the Berry-Esseen bound for the cumulative hazard estimator of F_x The cumulative hazard estimator can be represented as a U-statistic. By using the result in Helmers and van Zwet [6], we derive the Berry-esséen bound for this U-statistic. Then Berry-Esseen bounds for the distribution of the cumulative hazard estimator and the normal distribution and the distribution of the product-limit estimator and the normal distribution are obtained.     

Quantile process for left truncated and right censored data

In this paper, we consider the product-limit quantile estimator of an unknown quantile function when the data are subject to random left truncation and right censorship. This is a parallel problem to the estimation of the unknown distribution function by the product-limit estimator under the same model. Simultaneous strong Gaussian approximations of the product-limit process and product-limit quantile process are constructed with rate . A functional law of the iterated logarithm for the maximal deviation of the estimator from the estimand is derived from the construction. Work partially supported by NSC Grant 89-2118-M-259-011.     

Edgeworth expansion of the Studentized product-limit estimator for truncated and censored data

Based on random left truncated and right censored data we investigate the one-term Edgeworth expansion for the Studentized product-limit estimator, and show that the Edgeworth expansion is close to the exact distribution of the Studentized product-limit estimator with a remainder of On(su-1/2).     

随机左截断数据下乘积限估计的强逼近及其应用

本文建立了左截断数据下乘积限估计的强表示结果,其误差项的收敛速度达到重对数律。作为应用,推出了乘积限估计的重对数律和强逼近等深刻结果。     

左删失右截断数据的分位数的固定宽度序贯置信区间估计

基于左截断右删失数据下的乘积限估计构造了分位数固定宽度序贯置信区间及其估计，研究了序贯置信区间估计的渐近性质。作为副产品，获得了分位数估计近邻点的Bahadur表示定理。这个表示定理是推导分位数固定宽度序贯置信区间估计渐近性质的重要基础。同时，在文中，进行了一些计算机模拟试验，证明了左截断右删失数据下分位数估计的序贯方法是效的和精确的。     

截断与删失数据下的一个回归方法

对于截断与删失下的反映变量,我们提出了一类广义乘积限估计,并获得了它的弱收敛性.在回归分析中,利用这类广义乘积限估计来定义一种最小距离的参数估计,并获得了这种参数估计的相合性和渐近正态性.     

Bahadur-Kiefer theorems for the product-limit process

In the random censorship from the right model, strong and weak limit theorems for Bahadur-Kiefer type processes based on the product-limit estimator are established. The main theorm is sharp and may be considered as a final result as far as this type of research is concerned. As a consequence of this theorem a sharp uniform Bahadur representation for product-limit quantiles is obtained.     

The product-limit estimator and the bootstrap: Some asymptotic representations

Summary The product-limit estimator and its quantile process are represented as i.i.d. mean processes, with a remainder of ordern ^–3/4(logn)^3/4 a.s. Corresponding bootstrap versions of these representations are given, which can help one visualize how the bootstrap procedure operates in this set up.Research supported by NSF grants MCS-81-02341 and MCS 83-01082     

Asymptotic theorems for estimating the distribution function under random truncation

Representation theorem and local asymptotic minimax theorem are derived for nonparametric estimators of the distribution function on the basis of randomly truncated data. The convolution-type representation theorem asserts that the limiting process of any regular estimator of the distribution function is at least as dispersed as the limiting process of the product-limit estimator. The theorems are similar to those results for the complete data case due to Beran (1977, Ann. Statist., 5, 400–404) and for the censored data case due to Wellner (1982, Ann. Statist., 10, 595–602). Both likelihood and functional approaches are considered and the proofs rely on the method of Begun et al. (1983, Ann. Statist., 11, 432–452) with slight modifications.Division of Biostatistics, School of Public Health, Columbia Univ.     

Fixed Design Nonparametric Regression with Truncated and Censored Data

In this paper we consider a fixed design model in which the observations axe subject to left truncation and right censoring. A generalized product-limit estimator for the conditional distribution at a given covaxiate value is proposed, and an almost sure asymptotic representation of this estimator is established. We also obtain the rate of uniform consistency, weak convergence and a modulus of continuity for this estimator.Applications include trimmed mean and quantile function estimators.     

Asymptotic properties of conditional quantile estimator for censored dependent observations

In this paper, we establish strong uniform convergence and asymptotic normality of the conditional quantile estimator for the censorship model when the data exhibit some kind of dependence. It is assumed that the observations form a stationary α-mixing sequence. The strong uniform convergence in iid framework has recently been discussed by Ould-Saïd (Stat Probab Lett 76:579–586, 2006). As a by-product, we also obtain a uniform weak convergence rate for the product-limit estimator of the lifetime and censoring distributions under dependence, which is interesting independently.     

竞争风险场合PL型估计的强一致收敛性

本文构造了竞争风险场合分布函数的乘积极限（PL）型估计，运用经验过程的强逼近理论及Toylor展开方法，给出了PL型估计在全直线上的强一致收敛速度及其充分必要条件。     

删失数据的光滑乘积限估计的渐近性质

在删失数据的模型下，对于光滑未知的分布函数Ｆ０，文中提出了光滑化的方法去估计Ｆ０，得到了光滑ＰＬ估计Ｆｎ，并建立了Ｆｎ在Ｄ（－∞，Ｔ），Ｔ＜ＴＦ上的弱收敛和强相合的结果．同时也获得了光滑ＰＬ过程的强逼近和重对数律．     

STRONG REPRESENTATIONS OF THE SURVIVAL FUNCTION ESTIMATOR ON INCREASING SETS FOR TRUNCATED AND CENSORED DATA

1991MRSubjectClassification62G05,62E201IntroductionandtheMainResultsInthisarticlewestudysllrvivaldatawhicharesubjecttobothlefttrull(:ationandrightcensoring(LTRC).Morespecifically,let(X,T,Y)denoterandomvariableswhereXisthevariableofinterest,calledthelifetimevariable,withunknowndistributionfullctioll(d.f.)F;Tistherandomlefttruncatiolltiliiewitharbitraryd.f.G,andYistherandorl.1rightcensoringtirliewitharbitraryd.f.H.ItisassumedthatX,T,Yarerxlutuallyindependent.Intheran相似文献   

Uniform convergence of nonparametric regressions in competing risk models with right censoring

We consider, in the presence of covariates, non-independent competing risks that are subject to right censoring. We define a nonparametric estimator of the incident regression function through the generalized product-limit estimator of the conditional censorship distribution function. Under suitable conditions, we establish the almost sure uniform convergence of those estimators with an appropriate rate.     

左截断右删失数据分位差估计及其渐近性质

我们研究了左截断右删失数据分位差,基于左截断右删失数据乘积限构造了分位差的经验估计,同时克服经验估计的非光滑性,提出了分位数差的核光滑估计.利用经验过程理论推导出这两个估计的渐近偏差和渐近方差,并且在左截断右删失数据下研究了这两个分位差的大样本性质,获得分位差估计的相合性和渐近正态性.同时给出计算模拟以验证光滑分位差估计的表现,在均方损失的意义下模拟结果表明光滑估计比经验估计具有更好的性质.     

Tail product-limit process for truncated data with application to extreme value index estimation

A weighted Gaussian approximation to tail product-limit process for Pareto-like distributions of randomly right-truncated data is provided and a new consistent and asymptotically normal estimator of the extreme value index is introduced. A simulation study is carried out to evaluate the finite sample behavior of the proposed estimator and compare it to that recently proposed by Gardes and Stupfler (TEST 24, 207–227, 2015). Also, a new approach of estimating extreme quantiles, under random right truncation, is derived and applied to a real dataset of lifetimes of automobile brake pads.     

Statistical inference for right-censored data with nonignorable missing censoring indicators

We consider the statistical inference for right-censored data when censoring indicators are missing but nonignorable, and propose an adjusted imputation product-limit estimator. The proposed estimator is shown to be consistent and converges to a Gaussian process. Furthermore, we develop an empirical processbased testing method to check the MAR (missing at random) mechanism, and establish asymptotic properties for the proposed test statistic. To determine the critical value of the test, a consistent model-based bootstrap method is suggested. We conduct simulation studies to evaluate the numerical performance of the proposed method and compare it with existing methods. We also analyze a real data set from a breast cancer study for an illustration.     

Strong consistency properties of nonparametric estimators for randomly censored data,II: Estimation of density and failure rate

This article is Part II of a two-part study. Properties of the product-limit estimator established in the previous part [2] are now used to prove the strong consistency of some nonparametric density and failure rate estimators which can be used with randomly censored data.The third author's research was partly supported by the National Research Council of Canada.