Functional linear regression with points of impact
研究了函数线性回归的推广模型，其中存在未知数量的“冲击点”（即对响应变量有显著影响的离散观测时间点），并提出了估计冲击点数量和位置的方法，证明了可识别性和收敛速度。
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.