*(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 |