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, that are built upon functional principal components analysis, usually assume 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：我们提出了一种新的检验程序，用于在再生核希尔伯特空间框架下比较两组间的均值函数。该方法能够有效处理观测时间随机且采样稀疏不规则的功能数据。传统基于函数主成分分析的方法通常假设两组间具有同质协方差结构，然而这一假设在实际场景中往往难以验证。为消除对同质协方差结构的需求，我们首先在RKHS框架下推导了均值估计量的函数Bahadur表示，该表示自然导出了逐点渐近分布。进一步地，我们建立了均值估计量的弱收敛性，并据此构造了均值差异的检验统计量。该方法易于实现，且在控制第一类错误方面优于多种传统检验方法。通过大量模拟实验和两项实际应用案例，我们验证了该方法的有限样本性能。