  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.
