Asymptotic inference using functional principal component regression (FPCR) has long been considered difficult, largely because, upon any scalar scaling, the FPCR estimator fails to satisfy a central limit theorem, leading to the prevailing belief that it is unsuitable for direct statistical inference. In this paper, we upend this traditional viewpoint by establishing a new result: upon suitable operator scaling, valid Gaussian and bootstrap approximations hold for the FPCR estimator. We apply this surprising finding to hypothesis testing for the significance of the slope function in functional regression models and demonstrate the strong numerical performance of the resulting tests. While concise, our results yield powerful inferential tools for functional regression. We believe it paves the way for new lines of inferential methodology for more complex functional regression settings.

翻译：长期以来，函数主成分回归（FPCR）的渐近推断一直被认为具有挑战性，主要原因在于：经过任何标量缩放后，FPCR估计量均无法满足中心极限定理，这导致学界普遍认为其不适合直接用于统计推断。本文颠覆了这一传统观点，通过建立新的理论结果证明：经过适当的算子缩放后，对FPCR估计量可建立有效的高斯近似与自助法近似。我们将这一意外发现应用于函数回归模型中斜率函数显著性的假设检验，并展示了所构建检验方法的优异数值性能。尽管结论简洁，我们的研究结果为函数回归提供了强大的推断工具。我们相信这项工作将为更复杂的函数回归场景开辟新的推断方法论路径。