Truncated estimation in functional generalized linear regression models

Functional generalized linear models investigate the effect of functional predictors on a scalar response. An interesting case is when the functional predictor is thought to exert an influence on the conditional mean of the response only through its values up to a certain point in the domain. In the literature, models with this type of restriction on the functional effect have been termed truncated or historical regression models. A penalized likelihood estimator is formulated by combining a structured variable selection method with a localized B-spline expansion of the regression coefficient function. In addition to a smoothing penalty that is typical for functional regression, a nested group lasso penalty is also included which guarantees the sequential entering of B-splines and thus induces the desired truncation on the estimator. An optimization scheme is developed to compute the solution path efficiently when varying the truncation tuning parameter. The convergence rate of the coefficient function estimator and consistency of the truncation point estimator are given under suitable smoothness assumptions. The proposed method is demonstrated through simulations and an application involving the effects of blood pressure values in patients who suffered a spontaneous intracerebral hemorrhage.