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 cannot be asymptotically tractable. To attack this difficulty we propose a bias correction to achieve the standard "χ"-super-2-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 needed; when estimating nonparametric functions in the model, undersmoothing for ensuring √"n"-consistency 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 confidence 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. Copyright 2006 Royal Statistical Society.

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.KEY WORDS: Confidence band; Maximum empirical likelihood estimator.

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The empirical likelihood method is especially useful for constructing confidence 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 confidence 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 confidence regions (intervals).

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a b s t r a c tThe 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.

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a b s t r a c tThis paper focuses on the variable selections for semiparametric varying coefficient partially linear models when the covariates in the parametric and nonparametric components are all measured with errors. A bias-corrected variable selection procedure is proposed by combining basis function approximations with shrinkage estimations. With appropriate selection of the tuning parameters, the consistency of the variable selection procedure and the oracle property of the regularized estimators are established. A simulation study and a real data application are undertaken to evaluate the finite sample performance of the proposed method.