In this paper, we develop a {\em unified} framework for testing relevant hypotheses in functional time series. The proposed approach accommodates one-sample, two-sample, and change point problems for contaminated observations under arbitrary sampling schemes. Combining B-spline estimation with self-normalization, we construct nuisance-parameter-free tests that bypass auxiliary estimation of long-run covariance functions and measurement-error variance functions. We establish asymptotic validity by exploiting a sequential Gaussian approximation for dependent random vectors of moderately high dimension, which leads to a pivotal limiting distribution. We also provide sufficient conditions for the non-degeneracy of the self-normalizer and establish consistent decision rules. A key theoretical finding is that the proposed tests detect \(n^{-1/2}\)-local alternatives under arbitrary sampling frequencies. This uncovers a sparse-to-dense phase transition distinct from those typically observed in functional data analysis: while the sampling frequency affects the asymptotic variance, the detection rate remains \(n^{-1/2}\), even in sparsely sampled regimes. We further study multiple change point alternatives and extend the theory to settings where consistent change point estimates are available. We also discuss the choice of self-normalizers, including the recently developed range-adjusted self-normalizer. Extensive simulations support the theoretical results, and applications to the AU.SHF implied volatility and traffic volume datasets demonstrate the practical utility of the proposed methods.

翻译：本文发展了函数型时间序列中检验相关假设的统一框架。所提方法可处理任意抽样方案下受污染观测的单样本、双样本及变点问题。通过结合B样条估计与自标准化，我们构建了无需辅助估计长期协方差函数及测量误差方差函数的无冗余参数检验。通过利用适用于中等高维相依随机向量的序列高斯逼近，我们建立了渐近有效性并获得枢轴极限分布。同时给出了自标准化器非退化的充分条件，并建立了一致决策准则。关键理论发现是：所提检验在任意抽样频率下均可检测\(n^{-1/2}\)局部邻域备择假设。这揭示了不同于函数型数据分析中常见现象的“稀疏-密集”相变：抽样频率虽影响渐近方差，但即使在稀疏抽样场景下，检测速率仍保持为\(n^{-1/2}\)。我们进一步研究了多变点备择假设，并将理论推广至存在一致变点估计的情形。此外讨论了自标准化器的选择，包括新近发展的范围调整自标准化器。大量模拟验证了理论结果，对美国标准隐含波动率及交通流量数据集的应用展示了所提方法的实用价值。