Inference for functional linear models in the presence of heteroscedastic errors has received insufficient attention given its practical importance; in fact, even a central limit theorem has not been studied in this case. At issue, conditional mean estimates have complicated sampling distributions due to the infinite dimensional regressors, where truncation bias and scaling issues are compounded by non-constant variance under heteroscedasticity. As a foundation for distributional inference, we establish a central limit theorem for the estimated conditional mean under general dependent errors, and subsequently we develop a paired bootstrap method to provide better approximations of sampling distributions. The proposed paired bootstrap does not follow the standard bootstrap algorithm for finite dimensional regressors, as this version fails outside of a narrow window for implementation with functional regressors. The reason owes to a bias with functional regressors in a naive bootstrap construction. Our bootstrap proposal incorporates debiasing and thereby attains much broader validity and flexibility with truncation parameters for inference under heteroscedasticity; even when the naive approach may be valid, the proposed bootstrap method performs better numerically. The bootstrap is applied to construct confidence intervals for centered projections and for conducting hypothesis tests for the multiple conditional means. Our theoretical results on bootstrap consistency are demonstrated through simulation studies and also illustrated with a real data example.

翻译：鉴于其实际重要性，异方差误差存在下的函数线性模型推断尚未得到充分关注；事实上，甚至在此情况下的中心极限定理也未被研究过。问题的关键在于，由于无限维回归元的存在，条件均值估计量具有复杂的抽样分布，其中截断偏差与尺度问题在异方差条件下因方差非恒定而进一步加剧。作为分布推断的基础，我们首先在一般相依误差下建立了估计条件均值的中心极限定理，随后提出了一种配对Bootstrap方法以更好地逼近抽样分布。所提出的配对Bootstrap并不遵循有限维回归元的标准Bootstrap算法，因为该版本在函数型回归元场景下仅能在狭窄的参数窗口内有效实施。其原因在于朴素Bootstrap构造中函数型回归元带来的偏差。我们的Bootstrap方案通过引入去偏处理，从而在异方差推断中获得了关于截断参数的更广泛有效性与灵活性；即使在朴素方法可能有效的情况下，所提出的Bootstrap方法在数值表现上也更为优越。该Bootstrap方法被应用于构建中心化投影的置信区间以及对多重条件均值进行假设检验。我们关于Bootstrap一致性的理论结果通过模拟研究得到验证，并辅以实际数据案例加以说明。