Sparse estimation of historical functional linear models with a nested group bridge approach

The conventional historical functional linear model relates the current value of the functional response at time $t$ to all past values of the functional covariate up to time $t$. Motivated by situations where it is more reasonable to assume that only recent, instead of all, past values of the functional covariate have an impact on the functional response, in this work we investigate the historical functional linear model with an unknown forward time lag into the history. In addition to estimating the bivariate regression coefficient function, we also aim to identify the historical time lag from the data, which is important in many applications. To this end, we propose an estimation procedure that uses the finite element method to conform naturally to the trapezoidal domain of the bivariate coefficient function. We use a nested group bridge penalty to facilitate simultaneous estimation of the bivariate coefficient function and the historical lag, and show that our proposed estimators are consistent. We demonstrate this method of estimation in a real data example investigating the effect of muscle activation recorded via the noninvasive electromyography (EMG) method on lip acceleration during speech production. In addition, we examine the finite sample performance of our proposed method in comparison with the conventional approach to estimation via simulation studies.