We develop methodology for testing relevant hypotheses about functional time series in a tuning‐free way. Instead of testing for exact equality, e.g. for the equality of two mean functions from two independent time series, we propose to test the null hypothesis of no relevant deviation. In the two‐sample problem this means that an $L^2$‐distance between the two mean functions is smaller than a prespecified threshold. For such hypotheses self‐normalization, which was introduced in 2010 by Shao, and Shao and Zhang and is commonly used to avoid the estimation of nuisance parameters, is not directly applicable. We develop new self‐normalized procedures for testing relevant hypotheses in the one‐sample, two‐sample and change point problem and investigate their asymptotic properties. Finite sample properties of the tests proposed are illustrated by means of a simulation study and data examples. Our main focus is on functional time series, but extensions to other settings are also briefly discussed.

中文翻译：

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通过自我归一化检验功能时间序列中的相关假设

我们开发了用于以无调节方式测试有关功能时间序列的相关假设的方法。代替检验精确相等性，例如检验来自两个独立时间序列的两个均值函数的相等性，我们建议检验无相关偏差的零假设。在两样本问题中，这意味着${\mathrm{大号}}^2$-两个均值函数之间的距离小于预定阈值。对于这种假设，Shao，Shao和Zhang于2010年引入的自归一化方法通常用于避免干扰参数的估计，不能直接应用。我们开发了一种新的自规范化程序来测试一样本，二样本和变更点问题中的相关假设，并研究它们的渐近性质。通过模拟研究和数据示例说明了建议的测试的有限样本属性。我们主要关注功能时间序列，但也简要讨论了对其他设置的扩展。