We investigate asymptotic inference in a linear regression model where both response and regressors are functions, using an estimator based on functional principal components analysis. Although this approach is widely used in functional data analysis, there remains significant room for developing its asymptotic properties for function-on-function regression. Our study targets the mean response at a new regressor with two primary aims. First, we refine the existing central limit theorem by relaxing certain technical conditions, which include generalizing the scaling factor, resulting in incorporating a broader class of random functions beyond those having scores with independence or finite higher moments. Second, we introduce a residual bootstrap method that enhances the calibration of various confidence sets for quantities related to mean response, while its consistency is rigorously verified. Numerical studies compare the finite sample performance of both asymptotic and bootstrap approaches, demonstrating higher accuracy of the latter. To illustrate bootstrap inference for mean response, we apply it to the Canadian weather dataset.

翻译：本文在线性回归模型中研究渐近推断，其中响应变量与回归变量均为函数，采用基于函数主成分分析的估计量。尽管该方法在函数数据分析中广泛应用，但其在函数对函数回归中的渐近性质仍有较大发展空间。本研究聚焦于新回归变量处的均值响应，主要目标有二：首先，通过放宽部分技术条件（包括推广缩放因子）改进现有中心极限定理，从而纳入更广泛的随机函数类，超越仅具有独立得分或有限高阶矩的情形；其次，提出一种残差自助法以增强均值响应相关量值的各类置信集校准能力，并严格验证其相合性。数值研究比较了渐近方法与自助法在有限样本下的表现，证明后者具有更高精度。为说明均值响应的自助法推断，我们将其应用于加拿大气象数据集。