  We study a scalar-on-function historical linear regression model which
assumes that the functional predictor does not influence the response when the
time passes a certain cutoff point. We approach this problem from the
perspective of locally sparse modeling, where a function is locally sparse if
it is zero on a substantial portion of its defining domain. In the historical
linear model, the slope function is exactly a locally sparse function that is
zero beyond the cutoff time. A locally sparse estimate then gives rise to an
estimate of the cutoff time. We propose a nested group bridge penalty that is
able to specifically shrink the tail of a function. Combined with the B-spline
basis expansion and penalized least squares, the nested group bridge approach
can identify the cutoff time and produce a smooth estimate of the slope
function simultaneously. The proposed locally sparse estimator is shown to be
consistent, while its numerical performance is illustrated by simulation
studies. The proposed method is demonstrated with an application of determining
the effect of the past engine acceleration on the current particulate matter
emission.
