TY - JOUR
T1 - Empirical likelihood confidence regions in a partially linear single-index model
AU - ZHU, Lixing
AU - Xue, Liugen
N1 - Lixing Zhu’s research was supported by grant HKBU7060/04P from the Research Grants Council of Hong Kong, Hong Kong, China. Liugen Xue was supported by the National Natural Science Foundation of China (grant 10571008), the Natural Science Foundation of Beijing City of China (grant 1042002) and the Science and Technology Development Project of the Education Committee of Beijing City (grant KM200510005009).
PY - 2006/6
Y1 - 2006/6
N2 - 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 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 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.
AB - 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 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 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.
KW - Confidence region
KW - Coverage probability
KW - Empirical likelihood
KW - Partially linear single-index models
KW - X-distribution
UR - http://www.scopus.com/inward/record.url?scp=33646709168&partnerID=8YFLogxK
U2 - 10.1111/j.1467-9868.2006.00556.x
DO - 10.1111/j.1467-9868.2006.00556.x
M3 - Journal article
AN - SCOPUS:33646709168
SN - 1369-7412
VL - 68
SP - 549
EP - 570
JO - Journal of the Royal Statistical Society. Series B: Statistical Methodology
JF - Journal of the Royal Statistical Society. Series B: Statistical Methodology
IS - 3
ER -