Locally polynomial Hilbertian additive regression

Abstract

In this paper a new additive regression technique is developed for response variables that take values in general Hilbert spaces. The proposed method is based on the idea of smooth backfitting that has been developed mainly for real-valued responses. The local polynomial smoothing device is adopted, which renders various advantages of the technique evidenced in the classical univariate kernel regression with real-valued responses. It is demonstrated that the new technique eliminates many limitations which existing methods are subject to. In contrast to the existing techniques, the proposed approach is equipped with the estimation of the derivatives as well as the regression function itself, and provides options to make the estimated regression function free from boundary effects and possess oracle properties. A comprehensive theory is presented for the proposed method, which includes the rates of convergence in various modes and the asymptotic distributions of the estimators. The efficiency of the proposed method is also demonstrated via simulation study and is illustrated through real data applications.