A semiparametric regression model for longitudinal data is considered. The empirical like-lihood method is used to estimate the regression coefficients and the baseline function, and to construct confidence regions and intervals. It is proved that the maximum empirical likelihood estimator of the regression coefficients achieves asymptotic efficiency and the estimator of the baseline function attains asymptotic normality when a bias correction is made. Two calibrated empirical likelihood approaches to inference for the baseline function are developed.We propose a groupwise empirical likelihood procedure to handle the inter-series dependence for the longitudinal semiparametric regression model, and employ bias correction to construct the empirical likelihood ratio functions for the parameters of interest. This leads us to prove a nonparametric version of Wilks' theorem. Compared with methods based on normal approximations, the empirical likelihood does not require consistent estimators for the asymptotic variance and bias. A simulation compares the empirical likelihood and normal-based methods in terms of coverage accuracies and average areas/lengths of confidence regions/intervals.

In this article local empirical likelihood-based inference for a varying coefficient model with longitudinal data is investigated. First, we show that the naive empirical likelihood ratio is asymptotically standard chi-squared when undersmoothing is employed. The ratio is self-scale invariant and the plug-in estimate of the limiting variance is not needed. Second, to enhance the performance of the ratio, mean-corrected and residual-adjusted empirical likelihood ratios are recommended. The merit of these two bias corrections is that without undersmoothing, both also have standard chi-squared limits. Third, a maximum empirical likelihood estimator (MELE) of the time-varying coefficient is defined, the asymptotic equivalence to the weighted least-squares estimator (WLSE) is provided, and the asymptotic normality is shown. By the empirical likelihood ratios and the normal approximation of the MELE/WLSE, the confidence regions of the time-varying coefficients are constructed. Fourth, when some components are of particular interest, we suggest using mean-corrected and residual-adjusted partial empirical likelihood ratios to construct the confidence regions/intervals. In addition, we also consider the construction of the simultaneous and bootstrap confidence bands. A simulation study is undertaken to compare the empirical likelihood, the normal approximation, and the bootstrap methods in terms of coverage accuracies and average areas/widths of confidence regions/bands. An example in epidemiology is used for illustration.

The purpose of this article is to use an empirical likelihood method to study the construction of confidence intervals and regions for the parameters of interest in linear regression models with missing response data. A class of empirical likelihood ratios for the parameters of interest are defined such that any of our class of ratios is asymptotically chi-squared. Our approach is to directly calibrate the empirical log-likelihood ratio, and does not need multiplication by an adjustment factor for the original ratio. Also, a class of estimators for the parameters of interest is constructed, and the asymptotic distributions of the proposed estimators are obtained. Our results can be used directly to construct confidence intervals and regions for the parameters of interest. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths/areas of confidence intervals/regions. An example of a real data set is used for illustrating our methods.

In this article we study inference in parametric–nonparametric errors-in-covariables regression models using an empirical likelihood ap- proach based on validation data. It is shown that the asymptotic behavior of the proposed estimator depends on the ratio of the sizes of the primary sample and the validation sample; respectively. Unlike cases without measurement errors; the limit distribution of the estimator is no longer tractable and cannot be used for constructing condence regions. Monte Carlo approximations are employed to simulate the limit distribution. To increase the coverage accuracy of condence regions; two adjusted empirical likelihood estimators are recommended; which in the limit have a standard chi-squared distribution. A simulation study is carried out to compare the proposed methods with other existing methods. The new methods outperform the least squares method; and one of them works better than simulation–extrapolation (SIMEX) estimation; even when the restrictive model assumptions needed for SIMEX are satised. An application to a real dataset illustrates our new approach.

考虑纵向数据下部分线性模型，研究了回归系数和基准函数的经验似然推断，证明了所提出的经验对数似然比渐近于卡方分布，由此构造了相应兴趣参数的置信域和区间．此外，利用经验似然比函数得到了回归系数和基准函数的最大经验似然估计，并且证明了所得估计量的渐近正态性．模拟研究比较了经验似然与正态逼近方法的有限样本性质，并进行了案例分析．

Empirical-likelihood-based inference for the parameters in a partially linear single- index model is investigated. Unlike existing empirical likelihood procedures for other simpler models; if there is no bias correction the limit distribution of the empirical likelihood ratio can- not be asymptotically tractable. To attack this difculty we propose a bias correction to achieve the standard χ2 -limit. The bias-corrected empirical likelihood ratio shares some of the desired features of the existing least squares method: the estimation of the parameters is not p needed; when estimating nonparametric functions in the model; undersmoothing for ensuring n-con- sistency of the estimator of the parameters is avoided; the bias-corrected empirical likelihood is self-scale invariant and no plug-in estimator for the limiting variance is needed. Furthermore; since the index is of norm 1; we use this constraint as information to increase the accuracy of the condence regions (smaller regions at the same nominal level). As a by-product; our approach of using bias correction may also shed light on nonparametric estimation in model checking for other semiparametric regression models. A simulation study is carried out to assess the performance of the bias-corrected empirical likelihood. An application to a real data set is illustrated.

The empirical likelihood method is especially useful for constructing condence intervals or regions of the parameter of interest. This method has been extensively applied to linear regression and generalized linear regression models. In this paper; the empirical likelihood method for single-index regression models is studied. An estimated empirical log-likelihood approach to construct the condence region of the regression parameter is developed. An adjusted empirical log-likelihood ratio is proved to be asymptotically standard chi-square. A simulation study indicates that compared with a normal approximation-based approach; the proposed method described herein works better in terms of coverage probabilities and areas (lengths) of condence regions (intervals). 2005 Elsevier Inc. All rights reserved. AMS 1991 subject classication: primary 62G15; secondary 62E20

考虑带有协变量误差的非线性半参数模型，借助于核实数据，本文构造了未知参数的三种经验对数似然比统计量，证明了所提出的统计量具有渐近X2分布，此结果可以用来构造未知参数的置信域，另外，本文也构造了未知参数的最小二乘估计量，并证明了它的渐近性质，仅就置信域及其覆盖概率的大小方面，通过模拟研究比较了经验似然方法与最小二乘法的优劣，

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 suggested. It is proved that the proposed statistics are asymptotically standard chi-square under some suitable conditions; and hence can be used to construct the condence regions of the parameters. Our methods can also deal with the condence 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 condence regions; the proposed methods perform better than the least-squares method.

综合最小二乘法和局部线性光滑法给出了半参数回归模型中误差方差的估计量σ2n．在适当的条件下，证明了σ2n的Berrv-Esseen界可达到o(n-1/2).