Functional linear regression with points of impact

Abstract

The paper considers functional linear regression, where scalar responses $Y_1,\dots ,Y_n$ are modeled in dependence of i.i.d. random functions $X_1,\dots ,X_n$. 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 $X_1,\dots ,X_n$ 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.

Published at http://dx.doi.org/10.1214/15-AOS1323 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (9)

• Prediction in functional linear regression
• Methodology and convergence rates for functional linear regression
• Functional linear regression that's interpretable
• Smoothing splines estimators for functional linear regression
• Methodology and theory for partial least squares applied to functional data
• Adaptive Lasso for High Dimensional Regression and Gaussian Graphical Modeling
• Factor models and variable selection in high-dimensional regression analysis
• An RKHS formulation of the inverse regression dimension-reduction problem
• Fractals with point impact in functional linear regression

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