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, we investigate in this work the historical functional linear model with an unknown forward time lag into the history. Besides the common goal of estimating the bivariate regression coefficient function, we also aim to identify the historical time lag from the data, which is important in many applications. Tailored for this purpose, we propose an estimation procedure adopting the finite element method to conform naturally to the trapezoidal domain of the bivariate coefficient function. A nested group bridge penalty is developed to provide simultaneous estimation of the bivariate coefficient function and the historical lag. The method is demonstrated in a real data example investigating the effect of muscle activation recorded via the noninvasive electromyography (EMG) method on lip acceleration during speech production. The finite sample performance of our proposed method is examined via simulation studies in comparison with the conventional method.
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