Journal Article

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

Zhu, Lixing, Xue, Liugen

2006

Overview

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$\\chi^{2}-limit$. 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$\\sqrt{n}$-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.

• Publisher: Blackwell Publishing Ltd,Blackwell Publishers,Blackwell,Royal Statistical Society,Oxford University Press
• Subject: Bias / Confidence interval / Confidence region / Coverage probability / Distribution / Empirical likelihood / Empirical tests / Estimating techniques / Estimation / Estimation bias / Estimation methods / Estimators / Exact sciences and technology / Indexes / Inference / Least squares / Least squares method / Linear models / Linear regression / Mathematics / Maximum likelihood method / Nonparametric inference / Parametric inference / Partially linear single-index models / Polar regions / Probability / Probability and statistics / Regions / Sciences and techniques of general use / Scientific method / Simulation / Statistical methods / Statistical variance / Statistics / Studies / χ2-distribution