ABSTRACT We study additive function-on-function regression where the mean response at a particular time point depends on the time point itself, as well as the entire covariate trajectory. We develop a computationally efficient estimation methodology based on a novel combination of spline bases with an eigenbasis to represent the trivariate kernel function. We discuss prediction of a new response trajectory, propose an inference procedure that accounts for total variability in the predicted response curves, and construct pointwise prediction intervals. The estimation/inferential procedure accommodates realistic scenarios, such as correlated error structure as well as sparse and/or irregular designs. We investigate our methodology in finite sample size through simulations and two real data applications. Supplementary material for this article is available online.

中文翻译：

---

加法函数对函数回归

摘要我们研究了加性函数对函数回归，其中特定时间点的平均响应取决于时间点本身以及整个协变量轨迹。我们开发了一种基于样条基与特征基的新颖组合的计算高效估计方法，以表示三变量核函数。我们讨论了新响应轨迹的预测，提出了一个解释预测响应曲线总可变性的推理程序，并构建了逐点预测区间。估计/推理过程适应现实情况，例如相关错误结构以及稀疏和/或不规则设计。我们通过模拟和两个真实数据应用在有限样本量中研究我们的方法。