Open Access
February 2016 Functional linear regression with points of impact
Alois Kneip, Dominik Poß, Pascal Sarda
Ann. Statist. 44(1): 1-30 (February 2016). DOI: 10.1214/15-AOS1323


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.


Download Citation

Alois Kneip. Dominik Poß. Pascal Sarda. "Functional linear regression with points of impact." Ann. Statist. 44 (1) 1 - 30, February 2016.


Received: 1 July 2014; Revised: 1 February 2015; Published: February 2016
First available in Project Euclid: 10 December 2015

zbMATH: 1228.05123
MathSciNet: MR3449760
Digital Object Identifier: 10.1214/15-AOS1323

Primary: 62G08 , 62M99
Secondary: 62J05

Keywords: Functional linear regression , Model selection , Nonstandard asymptotics , Stochastic processes

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.44 • No. 1 • February 2016
Back to Top