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 (projection of the slope function) estimates have complicated sampling distributions due to the infinite dimensional regressors, which create truncation bias and scaling problems that are compounded by non-constant variance under heteroscedasticity. As a foundation for distributional inference, we establish a central limit theorem for the estimated projection under general dependent errors, and subsequently we develop a paired bootstrap method to approximate 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 projections and for conducting hypothesis tests for the slope function. Our theoretical results on bootstrap consistency are demonstrated through simulation studies and also illustrated with real data examples.

翻译：异方差误差存在时的函数线性模型推断因其实际重要性而尚未得到充分关注；事实上，该情形下甚至尚未研究过中心极限定理。问题在于，条件均值（斜率函数的投影）估计由于无限维回归量而具有复杂的抽样分布，这会产生截断偏差和尺度问题，而在异方差下非恒定方差又加剧了这些问题。作为分布推断的基础，我们建立了在一般相依误差下估计投影的中心极限定理，随后开发了配对Bootstrap方法来近似抽样分布。所提出的配对Bootstrap并不遵循有限维回归量的标准Bootstrap算法，因为该版本在函数回归量的实现窗口之外会失效。原因在于朴素Bootstrap构造中函数回归量存在偏差。我们的Bootstrap方案结合了去偏方法，从而在异方差推断中获得了更广泛的截断参数有效性和灵活性；即使朴素方法可能有效，所提出的Bootstrap方法在数值表现上也更优。该Bootstrap方法被应用于构建投影的置信区间以及进行斜率函数的假设检验。关于Bootstrap一致性的理论结果通过模拟研究得到验证，并辅以真实数据示例加以说明。