FUNCTIONAL LINEAR REGRESSION WITH POINTS OF IMPACT
成果类型：
Article
署名作者：
Kneip, Alois; Poss, Dominik; Sarda, Pascal
署名单位：
University of Bonn; University of Bonn; University of Bonn; Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Universite Federale Toulouse Midi-Pyrenees (ComUE); Institut National des Sciences Appliquees de Toulouse; Centre National de la Recherche Scientifique (CNRS)
刊物名称：
ANNALS OF STATISTICS
ISSN/ISSBN：
0090-5364
DOI：
10.1214/15-AOS1323
发表日期：
2016
页码：
1-30
关键词：
models
methodology
estimators
摘要：
The paper considers functional linear regression, where scalar responses Y1,..., Yn are modeled in dependence of i.i.d. random functions X1,..., Xn. 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 X1,..., Xn 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.