Functional regressors complicate inference in linear regression problems so that the bootstrap can play a useful role in quantifying uncertainty and calibrating intervals. The best bootstrap in practice, though, can depend on factors in the data as well as computational considerations and existing bootstraps can have limitations: residual bootstrap is computationally fast and simple but may fail when the errors are heterogeneous, while paired bootstrap applies more generally in functional linear regression at a cost of much higher computation. To bridge this gap, we develop a wild bootstrap method for functional linear regression, which is akin to a modified version of residual bootstrap but designed to have a wide scope of application like paired bootstrap, including to heteroscedastic errors. Its theoretical consistency is established and numerical studies suggest that wild bootstrap can provide accurate and computationally fast inference. Importantly, we also suggest a practical and effective approach of selecting truncation levels, specifically designed for mean response inference problems. The proposed bootstrap in functional linear regression is further illustrated through a weather data example, and an accompanying R package BTSinFLRM provides numerical implementations.

翻译：函数型回归变量使线性回归问题中的推断变得复杂，因此自助法在量化不确定性和校准区间方面可发挥重要作用。然而实际中最优自助法取决于数据特征与计算考量，现有自助法存在局限性：残差自助法计算快速简便，但当误差存在异质性时可能失效；配对自助法虽能更广泛适用于函数型线性回归，但计算成本显著增加。为弥补这一不足，我们提出适用于函数型线性回归的野自助法——该方法类似于残差自助法的改进版本，但具备与配对自助法同样广泛的适用范围（包括异方差误差）。我们建立了该方法的理论一致性，数值研究表明野自助法能提供精确且计算高效的推断。特别地，我们还针对均值响应推断问题，提出了一种实用有效的截断水平选择策略。通过天气数据实例进一步阐释了所提出的函数型线性回归自助法，并提供了配套的R语言软件包BTSinFLRM作为数值实现。