Functional linear regression with points of impact

TL;DR

This paper extends functional linear regression models by incorporating unknown discrete points of impact, proposing methods to identify these points and estimate model parameters with proven consistency and convergence rates.

Contribution

It introduces a novel generalized model with points of impact, providing a consistent estimation method and theoretical analysis of the model's identifiability and convergence properties.

Findings

Points of impact are identifiable under specific local variation conditions.

The proposed estimation method accurately determines the number and locations of impact points.

Convergence rates for impact point estimates and regression parameters are established.

Abstract

The paper considers functional linear regression, where scalar responses $Y_1,\ldots,Y_n$ are modeled in dependence of i.i.d. random functions $X_1,\ldots,X_n$. We study a generalization of the classical functional linear regression model. It is assumed that there exists an unknown number of "points of impact," that is, discrete observation times where the corresponding functional values possess significant influences on the response variable. In addition to estimating a functional slope parameter, the problem then is to determine the number and locations of points of impact as well as corresponding regression coefficients. Identifiability of the generalized model is considered in detail. It is shown that points of impact are identifiable if the underlying process generating $X_1,\ldots,X_n$ possesses "specific local variation." Examples are well-known processes like the Brownian motion, fractional Brownian motion or the Ornstein-Uhlenbeck process. The paper then proposes an easily implementable method for estimating the number and locations of points of impact. It is shown that this number can be estimated consistently. Furthermore, rates of convergence for location estimates, regression coefficients and the slope parameter are derived. Finally, some simulation results as well as a real data application are presented.