Partially functional linear regression with quadratic regularization

Partially functional regression, in which both functional predictor and multivariate predictors enter the model linearly, has been studied previously based on the functional principal component analysis (FPCA) approach. Here we consider an alternative estimation method based on Tikhonov regularization. Theoretically the regularization approach does not require the eigen-gap condition used in the functional PCA approach. The eventual goal of this study is to consider divide-and-conquer-type estimation of this model, which is suitable for distributed massive data. Convergence rates of the functional predictor and asymptotic normality of the multivariate predictors are derived for both the central estimator and the distributed estimator, and the asymptotic properties for the two are shown to be the same under appropriate conditions. Some Monte Carlo studies are carried out to illustrate the performances.