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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 χ 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.

62H15Multivariate hypothesis testing
62J05Linear regression
62G08Nonparametric regression
62E20Asymptotic distribution theory in statistics
62H10Multivariate distributions of statistics