Estimating Historical Functional Linear Models with a Nested Group Bridge Approach

TL;DR

This paper introduces a nested group bridge approach for estimating historical linear models, enabling simultaneous cutoff time detection and smooth slope function estimation in scalar-on-function regression.

Contribution

It proposes a novel nested group bridge penalty method that effectively identifies the cutoff time and estimates the slope function in historical linear models.

Findings

The estimator is shown to be consistent.

Simulation studies demonstrate good numerical performance.

Application illustrates the method's practical utility.

Abstract

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.