Unified theory of testing relevant hypotheses in functional time series
函数型时间序列实质性假设检验的统一理论AI 解读 ↓
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函数型时间序列实质性假设检验的统一理论
- 研究问题
- 在含测量污染且采样频率不一的函数型时间序列中，如何统一检验单样本、双样本及变点问题中的实质性差异？
- 方法与思路
- 作者结合 B 样条估计与自归一化，借助中等高维相依向量的序贯高斯逼近，构造无需另估长期协方差和测量误差方差的检验。
- 材料陈述的贡献
- 摘要称建立了枢轴极限分布、判决一致性及多变点扩展，并在所述条件下获得不随采样频率改变的 n^{-1/2} 局部检测率。
- 阅读关注点
- 应核对实质性差异阈值、采样与污染条件、自归一化分母非退化条件，以及稀疏采样下检测率不变所需的完整假设。
- 阅读边界
- 这是 arXiv 预印本，同行评审及期刊接收未核实；摘要未列出任意采样方案背后的正则条件，不能将检测率结论无条件推广。
原文摘要
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.