带有辅助信息的多元失效时间数据的估计标准部分似然方法

在许多实际研究中, 由于预算限制, 主协变量值只能对某一个有效集进行准确测量, 但同时对应此主协变量的辅助信息则对全部个体均可以观测. 利用这些辅助协变量的信息有助于提高统计研究的效率. 本文在基于共同基准危险率的边际模型框架下, 我们提出了一些统计推断方法来分析多元失效时间数据. 对于回归参数, 我们提出标准的估计部分似然方程来估计它, 同时也给出了累积基准危险率函数的Breslow 型估计. 得到的估计可以证明是相合的和渐近正态的. 利用模拟分析结果来表明了提出的方法在有限样本下的可行性.     

A class of weighted estimators for additive hazards model in case-cohort studies

Case-cohort sampling is a commonly used and efficient method for studying large cohorts. In many situations, some covariates are easily measured on all cohort subjects, and surrogate measurements of the expensive covariates also may be observed. In this paper, to make full use of the covariate data collected outside the case-cohort sample, we propose＇a class of weighted estimators with general time-varying weights for the additive hazards model, and the estimators are shown to be consistent and asymptotically normal. We also identify the estimator within this class that maximizes efficiency, and simulation studies show that the efficiency gains of the proposed estimator over the existing ones can be substantial in practical situations. A real example is provided.     

金融风险管理中VaR度量的光滑估计效率

本文研究了金融风险管理理论中风险价值(VaR)的非参数核光滑估计和经验估计的效率问题.对非独立的时间序列损失/收益样本,在均方误差(MSE)准则的意义下引入亏量的概念,亏量越大表明估计效率越低.并利用亏量对VaR模型的核光滑估计和基于样本分位数的经验估计进行了比较,在理论上证明了VaR模型的核光滑估计优于经验估计.同时,通过计算机模拟证实了理论获得的结论.本文还对国内沪深两市上的证券投资基金进行了实证分析,计算了样本基金的VaR风险度量的经验估计和核光滑估计,并计算了样本基金基于周收益率和VaR估计的风险调整收益(RAROC)值,以此对样本基金的业绩做出了有用的评价.     

Asymptotic properties of the maximum likelihood estimator for the proportional hazards model with doubly censored data

Doubly censored data, which include left as well as right censored observations, are frequently met in practice. Though estimation of the distribution function with doubly censored data has seen much study, relatively little is known about the inference of regression coefficients in the proportional hazards model for doubly censored data. In particular, theoretical properties of the maximum likelihood estimator of the regression coefficients in the proportional hazards model have not been proved yet. In this paper, we show the consistency and asymptotic normality of the maximum likelihood estimator and prove its semiparametric efficiency. The proposed methods are illustrated with simulation studies and analysis of an application from a medical study.     

Combining conditional and unconditional moment restrictions with missing responses

Many statistical models, e.g. regression models, can be viewed as conditional moment restrictions when distributional assumptions on the error term are not assumed. For such models, several estimators that achieve the semiparametric efficiency bound have been proposed. However, in many studies, auxiliary information is available as unconditional moment restrictions. Meanwhile, we also consider the presence of missing responses. We propose the combined empirical likelihood (CEL) estimator to incorporate such auxiliary information to improve the estimation efficiency of the conditional moment restriction models. We show that, when assuming responses are strongly ignorable missing at random, the CEL estimator achieves better efficiency than the previous estimators due to utilization of the auxiliary information. Based on the asymptotic property of the CEL estimator, we also develop Wilks’ type tests and corresponding confidence regions for the model parameter and the mean response. Since kernel smoothing is used, the CEL method may have difficulty for problems with high dimensional covariates. In such situations, we propose an instrumental variable-based empirical likelihood (IVEL) method to handle this problem. The merit of the CEL and IVEL are further illustrated through simulation studies.     

Additive-multiplicative hazards model for case-cohort studies with multiple disease outcomes

Case-cohort design is an efficient and economical design to study risk factors for diseases with expensive measurements, especially when the disease rate is low. When several diseases are of interest, multiple case-cohort design studies may be conducted using the same subcohort. To study the association between risk factors and each disease occurrence or death, we consider a general additive-multiplicative hazards model for case-cohort designs with multiple disease outcomes. We present an estimation procedure for the regression parameters of the additive-multiplicative hazards model, and show that the proposed estimator is consistent and asymptotically normal. Large sample approximation works well in finite sample studies in simulation. Finally, we apply the proposed method to a real data example for illustration.     

回归系数的混合估计与最小二乘估计的两种相对效率

本文研究了线性回归模型中,回归系数的混合估计与最小二乘估计的相对效率,利用矩阵的相关性质和运算,导出了两者之间两种新的相对效率的上下界.     

Kernel estimators of the ROC Curve with censored data

Receiver operating characteristic (ROC) curves are often used to study the two sample problem in medical studies. However, most data in medical studies are censored. Usually a natural estimator is based on the Kaplan-Meier estimator. In this paper we propose a smoothed estimator based on kernel techniques for the ROC curve with censored data. The large sample properties of the smoothed estimator are established. Moreover, deficiency is considered in order to compare the proposed smoothed estimator of the ROC curve with the empirical one based on Kaplan-Meier estimator. It is shown that the smoothed estimator outperforms the direct empirical estimator based on the Kaplan-Meier estimator under the criterion of deficiency. A simulation study is also conducted and a real data is analyzed.     

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In this article, we develop efficient robust method for estimation of mean and covariance simultaneously for longitudinal data in regression model. Based on Cholesky decomposition for the covariance matrix and rewriting the regression model, we propose a weighted least square estimator, in which the weights are estimated under generalized empirical likelihood framework. The proposed estimator obtains high efficiency from the close connection to empirical likelihood method, and achieves robustness by bounding the weighted sum of squared residuals. Simulation study shows that, compared to existing robust estimation methods for longitudinal data, the proposed estimator has relatively high efficiency and comparable robustness. In the end, the proposed method is used to analyse a real data set.     

One-stage and two-stage DEA estimation of the effects of contextual variables

Two-stage data envelopment analysis (2-DEA) is commonly used in productive efficiency analysis to estimate the effects of operational conditions and practices on performance. In this method the DEA efficiency estimates are regressed on contextual variables representing the operational conditions. We re-examine the statistical properties of the 2-DEA estimator, and find that it is statistically consistent under more general conditions than earlier studies assume. We further show that the finite sample bias of DEA in the first stage carries over to the second stage regression, causing bias in the estimated coefficients of the contextual variables. This bias is particularly severe when the contextual variables are correlated with inputs. To address this shortcoming, we apply the result that DEA can be formulated as a constrained special case of the convex nonparametric least squares (CNLS) regression. Applying the CNLS formulation, we develop a new semi-nonparametric one-stage estimator for the coefficients of the contextual variables that directly incorporates contextual variables to the standard DEA problem. The proposed method is hence referred to as one-stage DEA (1-DEA). Evidence from Monte Carlo simulations suggests that the new 1-DEA estimator performs systematically better than the conventional 2-DEA estimator both in deterministic and noisy scenarios.     

一类基于秩次的稳健线性回归估计与诊断方法

线性回归模型的误差项不服从正态分布或存在多个离群点时,可以将残差秩次的某些函数作为权重引入估计模型来减少离群点的不良影响。本文从参数估计、稳健性质、回归诊断等方面对基于残差秩次的一类稳健回归方法进行介绍.通过模拟研究和实例分析表明,R和GR估计是一种估计效率较高的稳健回归方法,其中GR估计可同时避免X与Y空间离群点,而高失效点HBR估计可通过控制某个参数在稳健性与估计效率之间进行折衷.     

Asymptotic Analysis of the Jittering Kernel Density Estimator

Jittering estimators are nonparametric function estimators for mixed data. They extend arbitrary estimators from the continuous setting by adding random noise to discrete variables. We give an in-depth analysis of the jittering kernel density estimator, which reveals several appealing properties. The estimator is strongly consistent, asymptotically normal, and unbiased for discrete variables. It converges at minimax-optimal rates, which are established as a by-product of our analysis. To understand the effect of adding noise, we further study its asymptotic efficiency and finite sample bias in the univariate discrete case. Simulations show that the estimator is competitive on finite samples. The analysis suggests that similar properties can be expected for other jittering estimators.     

A POSTERIORI ERROR ESTIMATE OF OPTIMAL CONTROL PROBLEM OF PDE WITH INTEGRAL CONSTRAINT FOR STATE

In this paper, we study adaptive finite element discretization schemes for an optimal control problem governed by elliptic PDE with an integral constraint for the state. We derive the equivalent a posteriori error estimator for the finite element approximation, which particularly suits adaptive multi-meshes to capture different singularities of the control and the state. Numerical examples are presented to demonstrate the efficiency of a posteriori error estimator and to confirm the theoretical results.     

Reduced‐bias kernel estimators of a positive extreme value index

In this paper, we deal with the semi‐parametric estimation of the extreme value index, an important parameter in extreme value analysis. It is well known that many classic estimators, such as the Hill estimator, reveal a strong bias. This problem motivated the study of two classes of kernel estimators. Those classes generalize the classical Hill estimator and have a tuning parameter that enables us to modify the asymptotic mean squared error and eventually to improve their efficiency. Since the improvement in efficiency is not very expressive, we also study new reduced bias estimators based on the two classes of kernel statistics. Under suitable conditions, we prove their asymptotic normality. Moreover, an asymptotic comparison, at optimal levels, shows that the new classes of reduced bias estimators are more efficient than other reduced bias estimator from the literature. An illustration of the finite sample behaviour of the kernel reduced‐bias estimators is also provided through the analysis of a data set in the field of insurance.     

Semiparametric methods for left-truncated and right-censored survival data with covariate measurement error

Many methods have been developed for analyzing survival data which are commonly right-censored. These methods, however, are challenged by complex features pertinent to the data collection as well as the nature of data themselves. Typically, biased samples caused by left-truncation (or length-biased sampling) and measurement error often accompany survival analysis. While such data frequently arise in practice, little work has been available to simultaneously address these features. In this paper, we explore valid inference methods for handling left-truncated and right-censored survival data with measurement error under the widely used Cox model. We first exploit a flexible estimator for the survival model parameters which does not require specification of the baseline hazard function. To improve the efficiency, we further develop an augmented nonparametric maximum likelihood estimator. We establish asymptotic results and examine the efficiency and robustness issues for the proposed estimators. The proposed methods enjoy appealing features that the distributions of the covariates and of the truncation times are left unspecified. Numerical studies are reported to assess the finite sample performance of the proposed methods.

     

Estimation for parameters of interest in random effects growth curve models

In this paper, we consider the general growth curve model with multivariate random effects covariance structure and provide a new simple estimator for the parameters of interest. This estimator is not only convenient for testing the hypothesis on the corresponding parameters, but also has higher efficiency than the least-square estimator and the improved two-stage estimator obtained by Rao under certain conditions. Moreover, we obtain the necessary and sufficient condition for the new estimator to be identical to the best linear unbiased estimator. Examples of its application are given.     

The a Posteriori Error Estimator of SDG Method for Variable Coefficients Time-Harmonic Maxwell's Equations

In this paper, we study the a posteriori error estimator of SDG method for variable coefficients time-harmonic Maxwell's equations. We propose two a posteriori error estimators, one is the recovery-type estimator, and the other is the residual-type estimator. We first propose the curl-recovery method for the staggered discontinuous Galerkin method (SDGM), and based on the super-convergence result of the postprocessed solution, an asymptotically exact error estimator is constructed. The residual-type a posteriori error estimator is also proposed, and it's reliability and effectiveness are proved for variable coefficients time-harmonic Maxwell's equations. The efficiency and robustness of the proposed estimators is demonstrated by the numerical experiments.     

Linearly interpolated FDH efficiency score for nonconvex frontiers

This paper addresses the problem of estimating the monotone boundary of a nonconvex set in a full nonparametric and multivariate setup. This is particularly useful in the context of productivity analysis where the efficient frontier is the locus of optimal production scenarios. Then efficiency scores are defined by the distance of a firm from this efficient boundary. In this setup, the free disposal hull (FDH) estimator has been extensively used due to its flexibility and because it allows nonconvex attainable production sets. However, the nonsmoothness and discontinuities of the FDH is a drawback for conducting inference in finite samples. In particular, it is shown that the bootstrap of the FDH has poor performances and so is not useful in practice. Our estimator, the LFDH, is a linearized version of the FDH, obtained by linear interpolation of appropriate FDH-efficient vertices. It offers a continuous, smooth version of the FDH. We provide an algorithm for computing the estimator, and we establish its asymptotic properties. We also provide an easy way to approximate its asymptotic sampling distribution. The latter could offer bias-corrected estimator and confidence intervals of the efficiency scores. In a Monte Carlo study, we show that these approximations work well even in moderate sample sizes and that our LFDH estimator outperforms, both in bias and in MSE, the original FDH estimator.     

关于回归模型的参数估计效率

本文讨论回归模型的参数估计效率。本文说明了现有线性回归模型的参数估计效率的下界与真实的参数估计效率在很多情况下相差较大,而且这种下界对于实测数据处理很难得到精确值。本文给出了估算参数估计效率的仿真方法。理论分析表明,该方法给出的参数估计效率的估计较现有的下界估计更合理;仿真和实算结果表明,对于一大类线性和非线性回归模型,该方法给出的回归模型的参数估计效率的估计更接近模型参数估计效率的真值。     

Semi-parametric efficiency bounds for regression models under response-selective sampling: the profile likelihood approach

We obtain an information bound for estimates of parameters in general regression models where data are collected under a variety of response-selective sampling schemes, together with a simple formula for the asymptotic variance of the semi-parametric maximum likelihood estimate. This is compared to the bound and the estimate is found to be fully efficient in a variety of settings. A small simulation study is reported to illustrate the small-sample efficiency of the semi-parametric estimator.