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．