Functional data analysis is becoming increasingly popular to study data from real-valued random functions. Nevertheless, there is a lack of multiple testing procedures for such data. These are particularly important in factorial designs to compare different groups or to infer factor effects. We propose a new class of testing procedures for arbitrary linear hypotheses in general factorial designs with functional data. Our methods allow global as well as multiple inference of both, univariate and multivariate mean functions without assuming particular error distributions nor homoscedasticity. That is, we allow for different structures of the covariance functions between groups. To this end, we use point-wise quadratic-form-type test functions that take potential heteroscedasticity into account. Taking the supremum over each test function, we define a class of local test statistics. We analyse their (joint) asymptotic behaviour and propose a resampling approach to approximate the limit distributions. The resulting global and multiple testing procedures are asymptotic valid under weak conditions and applicable in general functional MANOVA settings. We evaluate their small-sample performance in extensive simulations and finally illustrate their applicability by analysing a multivariate functional air pollution data set.

翻译：功能数据分析正日益成为研究实值随机函数数据的流行方法。然而，此类数据缺乏多重检验程序。这在因子设计中尤为重要，可用于比较不同组别或推断因子效应。我们提出了一类新的检验程序，适用于具有功能数据的一般因子设计中的任意线性假设。我们的方法允许对单变量和多变量均值函数进行全局推断和多重推断，且无需假设特定的误差分布或同方差性。也就是说，我们允许组间协方差函数具有不同的结构。为此，我们采用考虑潜在异方差性的逐点二次型检验函数。通过取每个检验函数的上确界，我们定义了一类局部检验统计量。我们分析了它们的（联合）渐近性质，并提出了一种重抽样方法来逼近极限分布。所得的全局和多重检验程序在弱条件下具有渐近有效性，并适用于一般功能多元方差分析场景。我们通过大量模拟评估了它们在小样本下的性能，最后通过分析一个多变量功能空气污染数据集来说明其适用性。