A unified approach to hypothesis testing is developed for scalar-on-function, function-on-function, function-on-scalar models and particularly mixed models that contain both functional and scalar predictors. In contrast with most existing methods that rest on the large-sample distributions of test statistics, the proposed method leverages the technique of bootstrapping max statistics and exploits the variance decay property that is an inherent feature of functional data, to improve the empirical power of tests especially when the sample size is limited or the signal is relatively weak. Theoretical guarantees on the validity and consistency of the proposed test are provided uniformly for a class of test statistics.

翻译：针对标量对函数、函数对函数、函数对标量模型，特别是包含函数型和标量预测变量的混合模型，本文提出了一种统一的假设检验方法。与大多数基于检验统计量大样本分布的现有方法不同，所提出的方法利用极大统计量的自助法技术，并充分发挥函数型数据固有的方差衰减特性，从而提升检验的经验功效，尤其是在样本量有限或信号较弱的情况下。本文统一为一类检验统计量提供了所提检验有效性和一致性的理论保证。