Generalized empirical likelihood inference in semiparametric regression model for longitudinal data

In this paper, we consider the semiparametric regression model for longitudinal data. Due to the correlation within groups, a generalized empirical log-likelihood ratio statistic for the unknown parameters in the model is suggested by introducing the working covariance matrix. It is proved that the proposed statistic is asymptotically standard chi-squared under some suitable conditions, and hence it can be used to construct the confidence regions of the parameters. A simulation study is conducted to compare the proposed method with the generalized least squares method in terms of coverage accuracy and average lengths of the confidence intervals.     

区间数据情形下线性模型的经验似然推断

§1Introduction Instatisticalapplications,weoftenencounterintervalcensoreddatawhenafailure timeYcannotbeobserved,butcanonlybedeterminedtolieinanintervalobtainedfroma sequenceofexaminationtimes.Forinstance,themaximumdosagewhichpatientscan endureisconcerned.LetYibethemaximumdosagewhichtheithpatientcanendure,Ui,j(j=1,2,...,k)bethedosagewhichthepatienthasbeentested.ItisobviousthatYiis unobservable.SupposetheithpatientisnormalwhenthedosageisUi,j,andhe(orshe)is abnormalwhenthedosageisUi,j+1.Then…     

Empirical likelihood confidence regions of the parameters in a partially linear single-index model

In this paper,a partially linear single-index model is investigated,and three empirical log-likelihood ratio statistics for the unknown parameters in the model are sug- gested.It is proved that the proposed statistics are asymptotically standard chi-square un- der some suitable conditions,and hence can be used to construct the confidence regions of the parameters.Our methods can also deal with the confidence region construction for the index in the pure single-index model.A simulation study indicates that,in terms of cov- erage probabilities and average areas of the confidence regions,the proposed methods perform better than the least-squares method.     

Empirical likelihood-based subset selection for partially linear autoregressive models

Based on the empirical likelihood method, the subset selection and hypothesis test for parameters in a partially linear autoregressive model are investigated. We show that the empirical log-likelihood ratio at the true parameters converges to the standard chi-square distribution. We then present the definitions of the empirical likelihood-based Bayes information criteria （EBIC） and Akaike information criteria （EAIC）. The results show that EBIC is consistent at selecting subset variables while EAIC is not. Simulation studies demonstrate that the proposed empirical likelihood confidence regions have better coverage probabilities than the least square method, while EBIC has a higher chance to select the true model than EAIC.     

响应变量存在缺失时部分线性模型的经验似然推断

考虑响应变量带有缺失的部分线性模型,采用借补的思想,研究了参数部分和非参数部分的经验似然推断,证明了所提出的经验对数似然比统计量依分布收敛到χ2分布,由此构造参数部分和函数部分的置信域和逐点置信区间.对参数部分,模拟比较了经验似然与正态逼近方法;对函数部分,模拟了函数的逐点置信区间.     

纵向数据部分线性模型的广义经验似然推断

考虑纵向数据部分线性模型,针对纵向数据个体内的相关性特点,通过引入估计的作业协方差矩阵,构造了模型中未知参数的三种经验对数似然比统计量.在适当条件下,证明了所提出的统计量依分布收敛于χ~2分布,所得结果可以构造未知参数的置信域.最后通过模拟研究对所提方法进行了说明.     

半参数变系数部分线性模型的统计推断

本文在多种复杂数据下, 研究一类半参数变系数部分线性模型的统计推断理论和方法. 首先在纵向数据和测量误差数据等复杂数据下, 研究半参数变系数部分线性模型的经验似然推断问题, 分别提出分组的和纠偏的经验似然方法. 该方法可以有效地处理纵向数据的组内相关性给构造经验似然比函数所带来的困难. 其次在测量误差数据和缺失数据等复杂数据下, 研究模型的变量选择问题, 分别提出一个“纠偏” 的和基于借补值的变量选择方法. 该变量选择方法可以同时选择参数分量及非参数分量中的重要变量, 并且变量选择与回归系数的估计同时进行. 通过选择适当的惩罚参数, 证明该变量选择方法可以相合地识别出真实模型, 并且所得的正则估计具有oracle 性质.     

Empirical likelihood analysis of longitudinal data involving within-subject correlation

In this paper we use profile empirical likelihood to construct confidence regions for regression coefficients in partially linear model with longitudinal data. The main contribution is that the within-subject correlation is considered to improve estimation efficiency. We suppose a semi-parametric structure for the covariances of observation errors in each subject and employ both the first order and the second order moment conditions of the observation errors to construct the estimating equations. Although there are nonparametric estimators, the empirical log-likelihood ratio statistic still tends to a standard ?? _p ² variable in distribution after the nuisance parameters are profiled away. A data simulation is also conducted.     

Empirical likelihood-based inferences for the area under the ROC curve with covariates

In the receiver operating characteristic (ROC) analysis,the area under the ROC curve (AUC) is a popular summary index of discriminatory accuracy of a diagnostic test.Incorporating covariates into ROC analysis can improve the diagnostic accuracy of the test.Regression model for the AUC is a tool to evaluate the effects of the covariates on the diagnostic accuracy.In this paper,empirical likelihood (EL) method is proposed for the AUC regression model.For the regression parameter vector,it can be shown that the asymptotic distribution of its EL ratio statistic is a weighted sum of independent chi-square distributions.Confidence regions are constructed for the parameter vector based on the newly developed empirical likelihood theorem,as well as for the covariate-specific AUC.Simulation studies were conducted to compare the relative performance of the proposed EL-based methods with the existing method in AUC regression.Finally,the proposed methods are illustrated with a real data set.     

Double penalized variable selection procedure for partially linear models with longitudinal data

Based on the double penalized estimation method,a new variable selection procedure is proposed for partially linear models with longitudinal data.The proposed procedure can avoid the effects of the nonparametric estimator on the variable selection for the parameters components.Under some regularity conditions,the rate of convergence and asymptotic normality of the resulting estimators are established.In addition,to improve efficiency for regression coefficients,the estimation of the working covariance matrix is involved in the proposed iterative algorithm.Some simulation studies are carried out to demonstrate that the proposed method performs well.     

Empirical likelihood-based dimension reduction inference for linear error-in-responses models with validation study

In this paper, linear errors-in-response models are considered in the presence of validation data on the responses. A semiparametric dimension reduction technique is employed to define an estimator of β with asymptotic normality, the estimated empirical loglikelihoods and the adjusted empirical loglikelihoods for the vector of regression coefficients and linear combinations of the regression coefficients, respectively. The estimated empirical log-likelihoods are shown to be asymptotically distributed as weighted sums of independent x12 and the adjusted empirical loglikelihoods are proved to be asymptotically distributed as standard chi-squares, respectively.     

Asymptotic properties in semiparametric partially linear regression models for functional data

We consider the semiparametric partially linear regression models with mean function XTβ + g(z), where X and z are functional data. The new estimators of β and g(z) are presented and some asymptotic results are given. The strong convergence rates of the proposed estimators are obtained. In our estimation, the observation number of each subject will be completely flexible. Some simulation study is conducted to investigate the finite sample performance of the proposed estimators.     

纵向数据下部分非线性模型的广义经验似然推断

本文研究了纵向数据下部分非线性模型中未知参数的置信域的构造.利用经验似然方法,构造了非线性函数中未知参数的广义对数经验似然比统计量,证明了其渐近于卡方分布.同时,得到了未知参数的最大经验似然估计,并证明了其渐近正态性.     

An empirical likelihood method in a partially linear single-index model with right censored data

Empirical-likelihood-based inference for the parameters in a partially linear single-index model with randomly censored data is investigated. We introduce an estimated empirical likelihood for the parameters using a synthetic data approach and show that its limiting distribution is a mixture of central chi-squared distribution. To attack this difficulty we propose an adjusted empirical likelihood to achieve the standard χ2-limit. Furthermore, since the index is of norm 1, we use this constraint to reduce the dimension of parameters, which increases the accuracy of the confidence regions. A simulation study is carried out to compare its finite-sample properties with the existing method. An application to a real data set is illustrated.     

Empirical likelihood inference in partially linear single-index models for longitudinal data

The empirical likelihood method is especially useful for constructing confidence intervals or regions of parameters of interest. Yet, the technique cannot be directly applied to partially linear single-index models for longitudinal data due to the within-subject correlation. In this paper, a bias-corrected block empirical likelihood (BCBEL) method is suggested to study the models by accounting for the within-subject correlation. BCBEL shares some desired features: unlike any normal approximation based method for confidence region, the estimation of parameters with the iterative algorithm is avoided and a consistent estimator of the asymptotic covariance matrix is not needed. Because of bias correction, the BCBEL ratio is asymptotically chi-squared, and hence it can be directly used to construct confidence regions of the parameters without any extra Monte Carlo approximation that is needed when bias correction is not applied. The proposed method can naturally be applied to deal with pure single-index models and partially linear models for longitudinal data. Some simulation studies are carried out and an example in epidemiology is given for illustration.     

半参数变系数异方差部分线性模型下的经验似然

Consider the semiparametric varying-coefficient heteroscedastic partially linear model Y i = Xτiβ + Zτiα(Ti) + σiei,1 ≤ i ≤ n,where σ 2 i = f(Ui),β is a p × 1 column vector of unknown parameter,(Xi,Zi,Ti,Ui) are random design points,Y i are the response variables,α(·) is a q-dimensional vector of unknown functions,e i are random errors.For both cases that f(·) is known and unknown,we propose the empirical log-likelihood ratio statistics for the parameter β.For each case,a nonparametric version of Wilks’ theorem is derived.The results are then used to construct confidence regions of the parameter.Simulation studies are carried out to assess the performance of the empirical likelihood method.     

纵向数据单指标模型中参数的经验似然置信域

考虑纵向数据单指标模型,针对纵向数据组间独立的特点,提出了模型中未知参数的三种经验对数似然比统计量.在适当条件下,证明了所提出的统计量依分布收敛于x~2分布,所得结果可以构造未知参数的置信域.进一步证明了所提出的纠偏的经验对数似然比有许多优良的性质.通过模拟研究对所提方法进行了说明.     

纵向部分线性模型的分块经验似然的有效推断(英文)

本文考虑了纵向数据的部分线性模型.在考虑个体内部相关性的情况下,研究了回归系数的经验似然推断.对于任意的工作协方差矩阵,所给的经验似然比统计量服从渐近卡方分布,由此可以构造回归参数的置信域.模拟结果表明,在正确指定个体相关结构的情况下,推断的效率会显著的提高.最后,给出了案例分析.     

双重广义线性模型的经验似然推断

基于截面经验似然方法,将双重广义线性模型的拟似然估计方程作为截面经验似然比函数的约束条件,构造了均值模型和散度模型未知参数的置信区间.最后通过数据模拟,将该方法与正态逼近方法比较,说明了该方法是有效和可行的.     

右删失数据下非线性回归模型的经验似然推断

考虑随机右删失数据下非线性回归模型,提出了模型中未知参数的调整的经验对数似然比统计量.在一定的条件下,证明了.所提出的的统计量具有渐近χ~2分布,由此结果构造了兴趣参数的置信域.通过模拟研究,对经典的经验似然、调整的经验似然和非线性最小二乘方法在有限样本下进行了比较,并对氯离子浓度试验数据进行了分析.