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This paper considers empirical likelihood inference for a high-dimensional partially functional linear model. An empirical log-likelihood ratio statistic is constructed for the regression coefficients of non-functional predictors and proved to be asymptotically normally distributed under some regularity conditions. Moreover, maximum empirical likelihood estimators of the regression coefficients of non-functional predictors are proposed and their asymptotic properties are obtained. Simulation studies are conducted to demonstrate the performance of the proposed procedure and a real data set is analyzed for illustration.

In this paper, we propose time-varying coefficient autoregressive models (TVARMs) of $p$th order as a solution for capturing the nonlinear structure of multiple explanatory variables. The stationary $\alpha$-mixing sequence of autoregressive variables is considered. We apply empirical likelihood approach to investigate confidence regions of the autoregressive coefficients of the TVARMs, and establish the nonparametric version of Wilks’ theorem of the proposed empirical log-likelihood ratio. Further, the maximum empirical likelihood estimator of the autoregressive coefficients and its asymptotic distribution are also obtained. Besides, to determine the order of autoregression, penalized empirical likelihood approach is developed with the help of smoothly clipped absolute deviation penalty and the resultant penalized estimators have the oracle property. Simulation studies and an analysis of monthly Chinese price index dataset are used to demonstrate the effectiveness of our proposed methods.