Empirical Likelihood Confidence Regions in a Partially Linear Single-Index Model

Abstract

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.

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

“…At present, the estimation problems of the single-index model have been widely studied. For instance, Ichimura [1] studied the properties of semiparametric least-squares estimators in general single-index model; Härdle and Tsybakov [2] developed the average derivative method to obtain the estimators of the index vector; Yu and Ruppert [3] proposed P-spline estimation for partially linear single-index models and studied asymptotic properties of the penalized least square estimators; Zhu and Xue [4] investigated empirical-likelihood-based inference for the parameters in a partially linear single-index model, they defined a bias-corrected empirical log-likelihood ratios and obtained the standard χ 2 distribution as its asymptotic distribution; Carroll et al [5] studied generalized partially linear single-index models; Xia et al [6] explored an extended version of a generalized partially linear single-index model.…”

A review on empirical likelihood methods for regression

“…Now, let Rn(β (r) ) be the EL ratio obtained by replacing the ξ i 's by η i 's. Then, Zhu and Xue (2006) showed that Wilks' theorem holds, i.e.…”

Empirical likelihood in some nonparametric and semiparametric models

“…The classical bias corrections in literature are used. Section 3 expatiates the results on the partially linear single-index model to describe the new bias correction proposed in Zhu and Xue (2006). Section 4 summarizes the results on the partial linear model with longitudinal data, and Section 5 presents the bias correction method for the varying coefficient model with longitudinal data.…”