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Profile likelihood inferences on semiparametric varying-coefficient partially linear models. (English) Zbl 1098.62077

Summary: Varying-coefficient partially linear models are frequently used in statistical modelling, but their estimation and inference have not been systematically studied. This paper proposes a profile least-squares technique for estimating the parametric component and studies the asymptotic normality of the profile least-squares estimator. The main focus is the examination of whether the generalized likelihood technique developed by J. Fan et al. [Ann. Stat. 29, No. 1, 153–193 (2001; Zbl 1029.62042)] is applicable to the testing problem for the parametric component of semiparametric models.

We introduce the profile likelihood ratio test and demonstrate that it follows an asymptotically ${\chi }^{2}$ distribution under the null hypothesis. This not only unveils a new Wilks type phenomenon, but also provides a simple and useful method for semiparametric inferences. In addition, the Wald statistic for semiparametric models is introduced and demonstrated to possess a sampling property similar to the profile likelihood ratio statistic. A new and simple bandwidth selection technique is proposed for semiparametric inferences on partially linear models and numerical examples are presented to illustrate the proposed methods.

##### MSC:
 62H15 Multivariate hypothesis testing 62J05 Linear regression 62G08 Nonparametric regression 62E20 Asymptotic distribution theory in statistics 62H10 Multivariate distributions of statistics
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