## The Annals of Statistics

### Functional linear regression with points of impact

#### 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.

#### Article information

Source
Ann. Statist., Volume 44, Number 1 (2016), 1-30.

Dates
Revised: February 2015
First available in Project Euclid: 10 December 2015

https://projecteuclid.org/euclid.aos/1449755955

Digital Object Identifier
doi:10.1214/15-AOS1323

Mathematical Reviews number (MathSciNet)
MR3449760

Zentralblatt MATH identifier
1228.05123

Subjects
Primary: 62G08: Nonparametric regression 62M99: None of the above, but in this section
Secondary: 62J05: Linear regression

#### Citation

Kneip, Alois; Poß, Dominik; Sarda, Pascal. Functional linear regression with points of impact. Ann. Statist. 44 (2016), no. 1, 1--30. doi:10.1214/15-AOS1323. https://projecteuclid.org/euclid.aos/1449755955

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#### Supplemental materials

• Supplement to “Functional linear regression with points of impact”. The supplementary document by Kneip, Poss and Sarda (2015) contains three Appendices. An application to NIR data can be found in Appendix A. In Appendix B, it is shown that the eigenfunctions of a Brownian motion satisfy assertion 2.5 in Theorem 2. Appendix C provides the proofs of Theorem 4 and Propositions 1 and 2.