Hazard Rate Estimation in Nonparametric Regression with Censored Data

Consider a regression model in which the responses are subject to random right censoring. In this model, Beran studied the nonparametric estimation of the conditional cumulative hazard function and the corresponding cumulative distribution function. The main idea is to use smoothing in the covariates. Here we study asymptotic properties of the corresponding hazard function estimator obtained by convolution smoothing of Beran's cumulative hazard estimator. We establish asymptotic expressions for the bias and the variance of the estimator, which together with an asymptotic representation lead to a weak convergence result. Also, the uniform strong consistency of the estimator is obtained.     

On estimation of survival function under random censoring model

We study an estimator of the survival function under the random censoring model. Bahadur-type representation of the estimator is obtained and asymptotic expression for its mean squared errors is given, which leads to the consistency and asymptotic normality of the estimator. A data-driven local bandwidth selection rule for the estimator is proposed. It is worth noting that the estimator is consistent at left boundary points, which contrasts with the cases of density and hazard rate estimation. A Monte Carlo comparison of different estimators is made and it appears that the proposed data-driven estimators have certain advantages over the common Kaplan-Meier estmator.     

响应变量删失时函数型部分线性分位数回归模型的估计

最近几年,函数型数据分析的理论和应用飞速发展.在许多实际应用里,响应变量往往存在随机右删失的情况.考虑利用函数型部分线性分位数回归模型来刻画函数型和标量预测量与右删失响应变量之间的关系.基于函数型主成分基函数来逼近未知的斜率函数,通过极小化逆概率加权分位数损失函数得到未知系数的估计量.文章的估计方法容易通过加权分位数回归程序实现.在一定的假设条件下,给出了有限维参数估计量的渐近正态性与斜率函数估计量的收敛速度.最后,通过模拟计算与应用实例证明了所提方法的有效性.     

Quantile regression for right-censored and length-biased data

Length-biased data arise in many important fields, including epidemiological cohort studies, cancer screening trials and labor economics. Analysis of such data has attracted much attention in the literature. In this paper we propose a quantile regression approach for analyzing right-censored and length-biased data. We derive an inverse probability weighted estimating equation corresponding to the quantile regression to correct the bias due to length-bias sampling and informative censoring. This method can easily handle informative censoring induced by length-biased sampling. This is an appealing feature of our proposed method since it is generally difficult to obtain unbiased estimates of risk factors in the presence of length-bias and informative censoring. We establish the consistency and asymptotic distribution of the proposed estimator using empirical process techniques. A resampling method is adopted to estimate the variance of the estimator. We conduct simulation studies to evaluate its finite sample performance and use a real data set to illustrate the application of the proposed method.     

Varying-coefficient model for the occurrence rate function of recurrent events

This article mainly considers the recurrent event process with independent censoring mechanism through a more flexible varying-coefficient model. The smoothing estimators for the varying-coefficient functions are also proposed via maximizing the kernel weight version of the log-partial likelihood function with respect to the coefficients at each time point. For the selection of appropriate bandwidths and the construction of confidence intervals, the consistent empirical smoothing estimators for the covariance functions of the estimators and a bias correction method are considered. As for the baseline effect function of recurrent events in the population, two different smoothing estimation methods are suggested and investigated. In this study, the asymptotic properties of the proposed smoothing estimators are derived. The finite sample properties of our methods are examined through a Monte Carlo simulation. Moreover, the procedures are applied to a recurrent sample of AIDS link to intravenous experiences (ALIVE) cohort study.     

Asymptotics for non-parametric likelihood estimation with doubly censored multivariate failure times

This paper considers non-parametric estimation of a multivariate failure time distribution function when only doubly censored data are available, which occurs in many situations such as epidemiological studies. In these situations, each of multivariate failure times of interest is defined as the elapsed time between an initial event and a subsequent event and the observations on both events can suffer censoring. As a consequence, the estimation of multivariate distribution is much more complicated than that for multivariate right- or interval-censored failure time data both theoretically and practically. For the problem, although several procedures have been proposed, they are only ad-hoc approaches as the asymptotic properties of the resulting estimates are basically unknown. We investigate both the consistency and the convergence rate of a commonly used non-parametric estimate and show that as the dimension of multivariate failure time increases or the number of censoring intervals of multivariate failure time decreases, the convergence rate for non-parametric estimate decreases, and is slower than that with multivariate singly right-censored or interval-censored data.     

Generalized MLE of a Joint Distribution Function with Multivariate Interval-Censored Data

We consider the problem of estimation of a joint distribution function of a multivariate random vector with interval-censored data. The generalized maximum likelihood estimator of the distribution function is studied and its consistency and asymptotic normality are established under the case 2 multivariate interval censorship model and discrete assumptions on the censoring random vectors.     

随机截断模型下生存函数的一个估计

本文讨论截断数据生存函数的估计问题。由于在寿命试验中截断分布Ｇ（ｙ）往往是人们５自己设计的，从类似Ｂｕｃｋｌｅｙ和Ｊａｍｅｓ处理期望的思想出发，文中给出了一个新的估计，并计算了它的期望和方差。     

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In this paper, we investigate a competing risks model based on exponentiated Weibull distribution under Type-I progressively hybrid censoring scheme. To estimate the unknown parameters and reliability function, the maximum likelihood estimators and asymptotic confidence intervals are derived. Since Bayesian posterior density functions cannot be given in closed forms, we adopt Markov chain Monte Carlo method to calculate approximate Bayes estimators and highest posterior density credible intervals. To illustrate the estimation methods, a simulation study is carried out with numerical results. It is concluded that the maximum likelihood estimation and Bayesian estimation can be used for statistical inference in competing risks model under Type-I progressively hybrid censoring scheme.     

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.     

Random weighting method for Cox’s proportional hazards model

Variance of parameter estimate in Cox’s proportional hazards model is based on asymptotic variance. When sample size is small, variance can be estimated by bootstrap method. However, if censoring rate in a survival data set is high, bootstrap method may fail to work properly. This is because bootstrap samples may be even more heavily censored due to repeated sampling of the censored observations. This paper proposes a random weighting method for variance estimation and confidence interval estimation for proportional hazards model. This method, unlike the bootstrap method, does not lead to more severe censoring than the original sample does. Its large sample properties are studied and the consistency and asymptotic normality are proved under mild conditions. Simulation studies show that the random weighting method is not as sensitive to heavy censoring as bootstrap method is and can produce good variance estimates or confidence intervals.     

The multiple imputations based Kaplan–Meier estimator

We derive the asymptotic distribution of the multiple imputations-based Kaplan–Meier estimator from right censored data with missing censoring indicators. We perform theoretical and numerical comparison studies with a competing semiparametric survival function estimator. We also carry out numerical studies to assess the performance of the proposed estimator when there is model misspecification.     

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In this paper, a nonparametric method for reliability of the stress-strength model is proposed when the dependent stress variable and strength variable are subject to right censoring. The dependence between variables is measured by the common Farlie-Gumbel-Morgenstern copula function and Clayton copula function. Using the empirical process theory, consistency and asymptotic normality of the proposed estimator is established in this paper. The results of numerical simulation show that the proposed method performs well in the case of finite sample. The method proposed in this paper has a wide application prospect in practice.     

Weighted local polynomial estimations of a non-parametric function with censoring indicators missing at random and their applications

In this paper, we consider the weighted local polynomial calibration estimation and imputation estimation of a non-parametric function when the data are right censored and the censoring indicators are missing at random, and establish the asymptotic normality of these estimators. As their applications, we derive the weighted local linear calibration estimators and imputation estimations of the conditional distribution function, the conditional density function and the conditional quantile function, and investigate the asymptotic normality of these estimators. Finally, the simulation studies are conducted to illustrate the finite sample performance of the estimators.     

左截断双删失数据下可加可乘风险率模型的估计

在生存分析中,可加可乘风险率模型常用来研究协变量对初始事件和终止事件之间持续时间的影响效应。在本文中,我们考虑了在初始事件存在部分区间删失,同时终止事件存在左截断右删失的情形下,持续时间的可加可乘风险率模型的估计问题。我们提出了一个两阶段估计过程来估计模型的回归参数。并通过模拟分析验证了估计的大样本性质。最后利用该方法分析了恶性黑色素瘤手术治疗数据。     

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??In this paper, we concern with the estimation problem for the Pareto distribution based on progressive Type-II interval censoring with random removals. We discuss the maximum likelihood estimation of the model parameters. Then, we show the consistency and asymptotic normality of maximum likelihood estimators based on progressive Type-II interval censored sample.     

Asymptotics for a weighted least squares estimator of the disease onset distribution function for a survival-sacrifice model

In carcinogenicity experiments with animals where the tumor is not palpable it is common to observe only the time of death of the animal, the cause of death (the tumor or another independent cause, as sacrifice) and whether the tumor was present at the time of death. These last two indicator variables are evaluated after an autopsy. A weighted least squares estimator for the distribution function of the disease onset was proposed. Asymptotic properties of that estimator are established here. We demonstrate its strong uniform consistency. A minimax lower bound for the estimation of the disease onset distribution is obtained, as well as the local asymptotic distribution for their estimator.     

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??In survival analysis, most existing approaches for analysing right-censored failure time data assume that the censoring time is independent of the failure time. However, investigators often face problems involving dependent censoring, i.e., failure time and censoring time are possibly dependent and they may be censored one another, especially in clinical trials. Without accounting for such dependence, survival distributions cannot be estimated consistently. Numerous attempts to model this dependence have been made. Among them, copula models are of particular interest because of their simple structure. Proportional hazard model analysis for informative right-censored data has been discussed in this paper. An Archimedean copula is assumed for the joint distribution function of failure time and censoring time variables. Under the conditions of identifiability of the parameter of the Archimedean copula, the maximum likelihood estimators of the parameter of Archimedean copula, the parameters and the cumulative hazard function of PH model are worked out. Extensive simulation studies show that the feasibility of the proposed method and the consistency of the estimators.     

Statistical Inference for the Rayleigh distribution under progressively Type-II censoring with binomial removal

This paper takes into account the estimation for the unknown parameter of the Rayleigh distribution under Type II progressive censoring with binomial removals, where the number of units removed at each failure time follows a binomial distribution. Maximum likelihood and Bayes procedure are used to derive both point and interval estimates of the parameters involved in the model. The expected termination point to complete the censoring test is computed and analyzed under binomial censoring scheme. Numerical examples are given to illustrate the approach by means of Monte Carlo simulation. A real life data set is used for illustrative purposes in conclusion.     

删失数据中删失指标随机缺失下回归函数的非参数估计

本文在删失数据中删失指标随机缺失的情况下,运用非参数方法给出了回归函数的两种估计量,给出了估计量的一致收敛速度以及渐近分布,并进一步通过数值模拟验证了所提方法在有限样本下的性质.