Semi-functional varying coefficient mode-based regression

We propose estimating semi-functional varying coefficient regression based on the mode value through a kernel objective function, where the bandwidth included is treated as a tuning parameter to achieve efficiency and robustness. For estimation, functional principal component basis functions are utilized to approximate the slope function and functional predictor variable, while B-spline functions are employed to approximate the varying coefficient component. Under mild regularity conditions, the convergence rates of the resulting estimators for the unknown slope function and varying coefficient are established under various cases. To numerically estimate the proposed model, we recommend employing a computationally efficient mode expectation–maximization algorithm with the aid of a Gaussian kernel. The tuning parameters are selected using the mode-based Bayesian information criterion and cross-validation procedures. Built upon the generalized likelihood technique, we further develop a goodness-of-fit test to assess the constancy of varying coefficient functions and put forward a wild bootstrap procedure for estimating the corresponding critical values. The finite sample performance of the developed estimators is illustrated through Monte Carlo simulations and real data analysis related to the Tecator data. The results produced by the propounded method are compared favorably with those obtained from alternative estimation techniques.