AI 中文总结
本文研究函数-函数线性回归模型，刻画其不可识别性，提出平滑样条估计量，揭示路径正则性与特征值衰减的权衡，验证区分并发与历史效应的实践重要性。
AI 中文摘要
我们研究一种函数-函数线性回归模型，其中时刻$t$的响应既依赖于预测变量的过去轨迹，也依赖于其并发值。该模型结合了$L^2$历史效应与点评估效应，且这两个系数函数并非自动可识别。我们刻画了由此产生的不可识别性，并证明当预测变量协方差算子的协方差特征函数不是逐点平方可和时，并发效应与历史效应可分别识别，这一温和且新颖的条件可避免并发点评估被$L^2$历史效应表征。基于预测变量过程的正交表示，我们为两个系数函数提出了平滑样条估计量，并建立了收敛速度。这些收敛速度揭示了路径正则性与特征值衰减间的有趣权衡：更平滑的预测变量轨迹会使历史效应估计量收敛更快，而更粗糙的轨迹则会使并发效应估计量收敛更快。模拟研究与两个实际数据应用证明了区分并发与历史效应的实践重要性。
英文摘要
We study a function-on-function linear regression model in which the response at time $t$ depends on both the past trajectory of a predictor and its concurrent value. The model combines an $L^2$-historical effect with a point-evaluation effect, and these two coefficient functions are not automatically identifiable. We characterize the resulting non-identifiability and show that the concurrent and historical effects are separately identifiable whenever the covariance eigenfunctions of the covariance operator of the predictor are not pointwise square-summable. This mild and novel condition prevents the concurrent point evaluation from being represented by an $L^2$-historical effect. Building on an orthogonalized representation of the predictor process, we propose a smoothing-spline estimator for both coefficient functions and establish consistency rates. The rates reveal an interesting trade-off between path regularity and eigenvalue decay: smoother predictor trajectories lead to faster convergence of the historical-effect estimator, whereas rougher trajectories lead to faster convergence of the concurrent-effect estimator. Simulation studies and two real-data applications demonstrate the practical importance of disentangling concurrent and historical effects.
