We propose a novel test procedure for comparing mean functions across two groups within the reproducing kernel Hilbert space (RKHS) framework. Our proposed method is adept at handling sparsely and irregularly sampled functional data when observation times are random for each subject. Conventional approaches, which are built upon functional principal components analysis, usually assume a homogeneous covariance structure across groups. Nonetheless, justifying this assumption in real-world scenarios can be challenging. To eliminate the need for a homogeneous covariance structure, we first develop the functional Bahadur representation for the mean estimator under the RKHS framework; this representation naturally leads to the desirable pointwise limiting distributions. Moreover, we establish weak convergence for the mean estimator, allowing us to construct a test statistic for the mean difference. Our method is easily implementable and outperforms some conventional tests in controlling type I errors across various settings. We demonstrate the finite sample performance of our approach through extensive simulations and two real-world applications.

翻译：我们在再生核希尔伯特空间（RKHS）框架下提出了一种新的检验程序，用于比较两组间的均值函数。该方法能够有效处理样本观测时间随机、数据稀疏且不规则的函数型数据。传统方法多基于函数主成分分析，通常假设组间具有同质协方差结构。然而，在实际场景中验证该假设往往具有挑战性。为消除对同质协方差结构的依赖，我们首先在RKHS框架下建立了均值估计量的函数Bahadur表示，该表示自然导出理想的逐点极限分布。进一步地，我们建立了均值估计量的弱收敛性，从而可构造均值差异的检验统计量。该方法易于实现，并且在多种设定下控制第一类错误方面优于若干传统检验。通过大量模拟实验和两项实际应用案例，我们验证了所提方法在有限样本下的表现。