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.     

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。     

Nonparametric estimators of the survival function with twice censored data

Patilea and Rolin (Ann Stat 34(2):925–938, 2006) proposed a product-limit estimator of the survival function for twice censored data. In this article, based on a modified self-consistent (MSC) approach, we propose an alternative estimator, the MSC estimator. The asymptotic properties of the MSC estimator are derived. A simulation study is conducted to compare the performance between the two estimators. Simulation results indicate that the MSC estimator outperforms the product-limit estimator and its advantage over the product-limit estimator can be very significant when right censoring is heavy.     

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.     

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).     

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.     

Bandwidth Selection in Density Estimation with Truncated and Censored Data

A plug-in type bandwidth selector is presented for density estimation with truncated and censored data. It is based on a representation of the MISE function obtained in the paper. Rate of convergence and limit distribution are derived for this selector. A bootstrap method is introduced to estimate the MISE whose minimizer is an alternative bandwidth selector. A simulation study was carried out to assess the behavior with small samples. This methodology is applied to a real-data problem consisting of reporting delay of AIDS cases. The almost sure representation of the product-limit estimator is a key tool in our proofs.     

The almost sure behavior of the oscillation modulus for PL-process and cumulative hazard process under random censorship

The local behavior of oscillation modulus of the product-limit (PL) process and the cumulative hazard process is investigated when the data are subjected to random censoring. Laws of the iterated logarithm of local oscillation modulus for the PL-process and the cumulative hazard process are established. Some of these results are applied to obtain the almost sure best rates of convergence for various types of density estimators as well as the Bahadur-Kiefer type process. Project supported in part by the National Natural Science Foundation of China (Grant No. 19701037).     

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.     

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   

One point stabilizers in almost simple sharp permutation groups

In this work we consider primitive sharp permutation groups of type ({0,l}, n). According to a previous result of the author, either the structure of such a group (as well as the associated action) is completely determined, or the group is almost simple. We investigate the viability of the almost simple case, develope additional restrictions on the structure of one point stabilizers, and give some indication of a strategy for a proof that there are no almost simple, primitive sharp permutation groups of type ({0l}n).     

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。     

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.     

Sharp constants for Moser‐Trudinger inequalities on spheres in complex space ℂn

The main results of this paper concern sharp constants for the Moser‐Trudinger inequalities on spheres in complex space ?ⁿ. We derive Moser‐Trudinger inequalities for smooth functions and holomorphic functions with different sharp constants (see Theorem 1.1). The sharp Moser‐Trudinger inequalities under consideration involve the complex tangential gradients for the functions and thus we have shown here such inequalities in the CR setting. Though there is a close connection in spirit between inequalities proven here on complex spheres and those on the Heisenberg group for functions with compact support in any finite domain proven earlier by the same authors [17], derivation of the sharp constants for Moser‐Trudinger inequalities on complex spheres are more complicated and difficult to obtain than on the Heisenberg group. Variants of Moser‐Onofri‐type inequalities are also given on complex spheres as applications of our sharp inequalities (see Theorems 1.2 and 1.3). One of the key ingredients in deriving the main theorems is a sharp representation formula for functions on the complex spheres in terms of complex tangential gradients (see Theorem 1.4). © 2004 Wiley Periodicals, Inc.     

Strong approximations of the quantile process of the product-limit estimator

The quantile process of the product-limit estimator (PL-quantile process) in the random censorship model from the right is studied via strong approximation methods. Some almost sure fluctuation properties of the said process are studied. Sections 3 and 4 contain strong approximations of the PL-quantile process by a generalized Kiefer process. The PL and PL-quantile processes by the same appropriate Kiefer process are approximated and it is demonstrated that this simultaneous approximation cannot be improved in general. Section 5 contains functional LIL for the PL-quantile process and also three methods of constructing confidence bands for theoretical quantiles in the random censorship model from the right.     

Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems

The paper is devoted to the problem of verification of accuracy of approximate solutions obtained in computer simulations. This problem is strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. A mathematical model embracing nonlinear elliptic variational problems is considered in this work. Based on functional type estimates developed on an abstract level, we present a general technology for constructing computable sharp upper bounds for the global error for various particular classes of elliptic problems. Here the global error is understood as a suitable energy type difference between the true and computed solutions. The estimates obtained are completely independent of the numerical technique used to obtain approximate solutions, and are sharp in the sense that they can be, in principle, made as close to the true error as resources of the used computer allow. The latter can be achieved by suitably tuning the auxiliary parameter functions, involved in the proposed upper error bounds, in the course of the calculations.     

Sharp Spectral Multipliers for a New Class of Grushin Type Operators

We describe weighted restriction (Plancherel) type estimates and sharp Hebisch-Müller-Stein type spectral multiplier result for a new class of Grushin type operators. We also discuss the optimal exponent for Bochner-Riesz summability in this setting.     

Weighted restriction type estimates for Grushin operators and application to spectral multipliers and Bochner–Riesz summability

We prove weighted restriction type estimates for Grushin operators. These estimates are then used to prove sharp spectral multiplier theorems as well as Bochner–Riesz summability results with sharp exponent.