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.     

Nonparametric inference in multiplicative intensity model by discrete time observation

This paper deals with nonparametric inference problems in the multiplicative intensity model for counting processes. We propose a Nelson–Aalen type estimator based on discrete observation. The functional asymptotic normality of the estimator is proved. The limit process is the same as that in the continuous observation case, thus the proposed estimator based on discrete observation has the same properties as the Nelson–Aalen estimator based on continuous observation. For example, the asymptotic efficiency of proposed estimator is valid based on less information than the continuous observation case. A Kaplan–Meier type estimator is also discussed. Nonparametric goodness of fit test is considered, and an asymptotically distribution free test is proposed.     

Multivariate density estimation using dimension reducing information and tail flattening transformations for truncated or censored data

This paper introduces a multivariate density estimator for truncated and censored data with special emphasis on extreme values based on survival analysis. A local constant density estimator is considered. We extend this estimator by means of tail flattening transformation, dimension reducing prior knowledge and a combination of both. The asymptotic theory is derived for the proposed estimators. It shows that the extensions might improve the performance of the density estimator when the transformation and the prior knowledge is not too far away from the true distribution. A simulation study shows that the density estimator based on tail flattening transformation and prior knowledge substantially outperforms the one without prior knowledge, and therefore confirms the asymptotic results. The proposed estimators are illustrated and compared in a data study of fire insurance claims.     

纵向数据非参数混合效应模型的一个局部不变估计

非参数核回归方法近年来已被用于纵向数据的分析(Lin和Carroll,2000)．一个颇具争议性的问题是在非参数核回归中是否需要考虑纵向数据间的相关性．Lin和Carroll (2000)证明了基于独立性(即忽略相关性)的核估计在一类核GEE估计量中是(渐近)最有效的．基于混合效应模型方法作者提出了一个不同的核估计类,它自然而有效地结合了纵向数据的相关结构．估计量达到了与Lin和Carroll的估计量相同的渐近有效性,且在有限样本情形下表现更好．由此方法可以很容易地获得对于总体和个体的非参数曲线估计．所提出的估计量具有较好的统计性质,且实施方便,从而对实际工作者具有较大的吸引力．     

A KERNEL-TYPE ESTIMATOR OF A QUANTILE FUNCTION UNDER RANDOMLY TRUNCATED DATA

A kernel-type estimator of the quantile function Q(p) = inf {t : F(t)≥p}, 0≤p≤1, is proposed based on the kernel smoother when the data are subjected to random truncation. The Bahadur-type representations of the kernel smooth estimator are established, and from Bahadur representations the authors can show that this estimator is strongly consistent, asymptotically normal, and weakly convergent.     

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.     

回归系数的广义根方估计及其模拟

文献[1,2]中提出了回归系数的根方估计~(k),当回归自变量间存在复共线关系时,~(k)较回归系数的最小二乘估计有所改善,本文将根方估计作一拓广,得出了回归系数的广义根方估计~(K),其中K为对角阵,文中证明了广义根方估计~(K)较~(k)能更有效地改善最小二乘估计,并给出了广义根方估计的显式解,在此基础上,提出了广义根方估计的显式解和一种确定k_i的方法。     

Locally Sparse Estimator for Functional Linear Regression Models

A new locally sparse (i.e., zero on some subregions) estimator for coefficient functions in functional linear regression models is developed based on a novel functional regularization technique called “fSCAD.” The nice shrinkage property of fSCAD allows the proposed estimator to locate null subregions of coefficient functions without over shrinking nonzero values of coefficient functions. Additionally, a roughness penalty is incorporated to control the roughness of the locally sparse estimator. Our method is theoretically sounder and computationally simpler than existing methods. Asymptotic analysis reveals that the proposed estimator is consistent and can identify null subregions with probability tending to one. Extensive simulations confirm the theoretical analysis and show excellent numerical performance of the proposed method. Practical merit of locally sparse modeling is demonstrated by two real applications. Supplemental materials for the article are available online.     

Gaining effciency via weighted estimators for mul-tivariate failure time data

Multivariate failure time data arise frequently in survival analysis.A commonly used tech-nique is the working independence estimator for marginal hazard models.Two natural questions are how to improve the effciency of the working independence estimator and how to identify the situations under which such an estimator has high statistical effciency.In this paper,three weighted estimators are proposed based on three different optimal criteria in terms of the asymptotic covariance of weighted estimators.Simpli...     

Nonparametric estimation of cumulative incidence functions for competing risks data with missing cause of failure

In this paper, we develop a fully nonparametric approach for the estimation of the cumulative incidence function with Missing At Random right-censored competing risks data. We obtain results on the pointwise asymptotic normality as well as the uniform convergence rate of the proposed nonparametric estimator. A simulation study that serves two purposes is provided. First, it illustrates in detail how to implement our proposed nonparametric estimator. Second, it facilitates a comparison of the nonparametric estimator to a parametric counterpart based on the estimator of Lu and Liang (2008). The simulation results are generally very encouraging.     

Semiparametric estimation in regression with missing covariates using single-index models

We investigate semiparametric estimation of regression coefficients through generalized estimating equations with single-index models when some covariates are missing at random. Existing popular semiparametric estimators may run into difficulties when some selection probabilities are small or the dimension of the covariates is not low. We propose a new simple parameter estimator using a kernel-assisted estimator for the augmentation by a single-index model without using the inverse of selection probabilities. We show that under certain conditions the proposed estimator is as efficient as the existing methods based on standard kernel smoothing, which are often practically infeasible in the case of multiple covariates. A simulation study and a real data example are presented to illustrate the proposed method. The numerical results show that the proposed estimator avoids some numerical issues caused by estimated small selection probabilities that are needed in other estimators.

     

Research on Inference of Candidate Database Web Surveys Based on Superpopulation Pseudo Design and the Combined Sample

How to solve the inference problem of candidate database web surveys is an urgent problem to be solved in the development of web survey. In order to solve this problem, the inference method of non-probability sampling based on superpopulation pseudo design and the combined sample is proposed. A superpopulation model is firstly built up to construct pseudo weights for a survey sample of the web candidate database. The estimator of the population mean is then computed according to the combined sample composed of the survey sample of the web candidate database and a probability sample. The variance estimator of the population mean estimator is lastly derived according to the variance estimation theory of the superpopulation model. The Bootstrap and Jackknife methods are also used to compute the variance estimator. And all these variance estimation methods are compared. The research results show that the population mean estimator based on superpopulation pseudo design and the combined sample is better, and has higher efficiency than the estimator only using the probability sample and the weighted estimator only using the survey sample of the web candidate database. The variance estimator computed by using the VM1, VM2 and VM3 method are relatively better.     

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??How to solve the inference problem of candidate database web surveys is an urgent problem to be solved in the development of web survey. In order to solve this problem, the inference method of non-probability sampling based on superpopulation pseudo design and the combined sample is proposed. A superpopulation model is firstly built up to construct pseudo weights for a survey sample of the web candidate database. The estimator of the population mean is then computed according to the combined sample composed of the survey sample of the web candidate database and a probability sample. The variance estimator of the population mean estimator is lastly derived according to the variance estimation theory of the superpopulation model. The Bootstrap and Jackknife methods are also used to compute the variance estimator. And all these variance estimation methods are compared. The research results show that the population mean estimator based on superpopulation pseudo design and the combined sample is better, and has higher efficiency than the estimator only using the probability sample and the weighted estimator only using the survey sample of the web candidate database. The variance estimator computed by using the VM1, VM2 and VM3 method are relatively better.     

药物剂量水平的识别及其渐近分布

本文在独立响应情况下考虑不会导致风险显著增加的最高剂量的估计问题,基于AIC(Akaike Inform ation Criterion)的思想提出了一个新的估计方法,给出了该估计的渐近分布。     

Consistent estimation of the dimensionality in sliced inverse regression

Several methods have been proposed in the literature in order to estimate the dimensionality in sliced inverse regression. Most of these methods are based on sequential tests for the nullity of the last eigenvalues of suitable operators. We first establish non consistency for estimators resulting from these methods. Then, we propose an estimator obtained by minimizing a suitable penalization of a statistic based on eigenvalues. A consistency property is established for this estimator and a simulation study is undertaken to evaluate its finite sample performance.     

Probability density estimation for survival data with censoring indicators missing at random

In this paper, some nonparametric approaches of density function estimation are developed when censoring indicators are missing at random. A conditional mean score based estimator and a mean score estimator are suggested, respectively. The two estimators are proved to be asymptotically normal and uniformly strongly consistent. The bandwidth selection problem is also discussed. A simulation study is conducted to compare finite-sample behaviors of the proposed estimators.     

Empirical likelihood estimation of discretely sampled processes of OU type

This paper presents an empirical likelihood estimation procedure for parameters of the discretely sampled process of Ornstein-Uhlenbeck type. The proposed procedure is based on the condi- tional characteristic function, and the maximum empirical likelihood estimator is proved to be consistent and asymptotically normal. Moreover, this estimator is shown to be asymptotically efficient under some mild conditions. When the background driving Lévy process is of type A or B, we show that the intensity parameter c...     

An a posteriori error estimator based on shifts for positive Hermitian eigenvalue problems

This work deals with an a posteriori error estimator for Hermitian positive eigenvalue problems. The proposed estimator is based on the residual and the definition of suitable shifts in the matrix spectrum. The mathematical properties (certification and sharpness) are investigated and some numerical experiments are proposed.     

Distribution estimation with smoothed auxiliary information

Distribution estimation is very important in order to make statistical inference for parameters or its functions based on this distribution.In this work we propose an estimator of the distribution of some variable with non-smooth auxiliary information,for example,a symmetric distribution of this variable.A smoothing technique is employed to handle the non-differentiable function.Hence,a distribution can be estimated based on smoothed auxiliary information.Asymptotic properties of the distribution estimator are derived and analyzed.The distribution estimators based on our method are found to be significantly efficient than the corresponding estimators without these auxiliary information.Some simulation studies are conducted to illustrate the finite sample performance of the proposed estimators.     

Jackknifing in partially linear regression models with serially correlated errors

In this paper jackknifing technique is examined for functions of the parametric component in a partially linear regression model with serially correlated errors. By deleting partial residuals a jackknife-type estimator is proposed. It is shown that the jackknife-type estimator and the usual semiparametric least-squares estimator (SLSE) are asymptotically equivalent. However, simulation shows that the former has smaller biases than the latter when the sample size is small or moderate. Moreover, since the errors are correlated, both the Tukey type and the delta type jackknife asymptotic variance estimators are not consistent. By introducing cross-product terms, a consistent estimator of the jackknife asymptotic variance is constructed and shown to be robust against heterogeneity of the error variances. In addition, simulation results show that confidence interval estimation based on the proposed jackknife estimator has better coverage probability than that based on the SLSE, even though the latter uses the information of the error structure, while the former does not.