In this paper, an F-type test is proposed for the functional association between a functional covariate and a scalar response in functional linear models (FLM). The proposed F test is based on the order statistics of functional principal components (fPCs), which is motivated by the phenomena that the power of the classical F test is sensitive to the choice of thresholds. Theoretical properties of the proposed test are established under both the null and alternative hypotheses, especially showing that it can detect the local alternative of order 𝑆1/4𝑛𝑛−1/2$S_n^{1/4}{n}^{-1/2}$$S_n^{1/4}{n}^{-1/2}$, that is, the fastest rate in the high-dimensional linear model, where n and S_n are the sample size and the number of fPCs, respectively. Extensive simulation studies are conducted to demonstrate the finite sample performance, showing advantages to the existing tests. Moreover, our procedure is applied to a real data analysis related to the diffusion tensor imaging tractography study.

中文翻译：

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函数线性模型中的 F 型检验

在本文中，针对功能线性模型 (FLM) 中功能协变量和标量响应之间的功能关联，提出了 F 型检验。所提出的 F 检验基于功能主成分 (fPC) 的阶数统计，其动机是经典 F 检验的功效对阈值的选择敏感。所提出的检验的理论性质是在零假设和备择假设下建立的，特别是表明它可以检测到顺序的局部替代𝑆1 / 4𝑛𝑛− 1 / 2${\mathrm{小号}}_n^{1/4}{n}^{-1/2}$$S_n^{1/4}{n}^{-1/2}$，即高维线性模型中的最快速率，其中n和S _n分别是样本大小和 fPC 的数量。进行了广泛的模拟研究以证明有限样本的性能，显示出现有测试的优势。此外，我们的程序适用于与扩散张量成像纤维束成像研究相关的真实数据分析。