Functional partial linear regression with quadratic regression for the multivariate predictor

In the last decades, functional semi-parametric models are widely studied such as functional additive model, functional partial linear model, functional index model, and so on. But some of the semi-parametric models suffer from the lower convergence rate for the non parametric parts (especially for the non parametric functional regression) and the assumption for the linear parts is too strict in practical. In this article, we propose a functional partial linear model corresponding to a scalar response, while the predictors contain both of function and multivariate predictor. We use a functional linear model to avoid the low convergence rate of the non parametric functional estimation, and for the multivariate predictor, we use a quadratic regression model to deal with the strict assumption for the linear model. To estimate this model, we first expand the functional predictor and functional regression parametric on the functional principal component basis, then, we estimate the functional coefficient by the least square method. For the theoretical studying, we will study the asymptotical normal distribution of the multivariate regression parameter and the specific convergence rate of the functional regression parameter for the dense functional observations. Furthermore, we illustrate the performance of the proposed method by simulation studies and one real data analysis.