Penalized Likelihood Functional Regression

This paper studies the generalized functional linear model with a scalar response and a functional predictor. The response given the functional predictor is assumed to come from the distribution of an exponential family. A penalized likelihood approach is proposed to estimate the unknown intercept and coefficient function in the model. Inference tools such as point-wise confidence intervals of the coefficient function and prediction intervals are derived. The minimax rate of convergence for the error in predicting the mean response is established. It is shown that the penalized likelihood estimator attains the optimal rate of convergence. Our simulations demonstrate a competitive performance against the existing approach. The method is further illustrated in the use of the DTI tractography to distinguish corpus callosum tracts with multiple sclerosis from normal tracts.