In this article, we study nonparametric inference problems in the context of multivariate or functional time series, including testing for goodness-of-fit, the presence of a change point in the marginal distribution, and the independence of two time series, among others. Most methodologies available in the existing literature address these problems by employing a bandwidth-dependent bootstrap or subsampling approach, which can be computationally expensive and/or sensitive to the choice of bandwidth. To address these limitations, we propose a novel class of kernel-based tests by embedding the data into a reproducing kernel Hilbert space, and construct test statistics using sample splitting, projection, and self-normalization (SN) techniques. Through a new conditioning technique, we demonstrate that our test statistics have pivotal limiting null distributions under absolute regularity and mild moment assumptions. We also analyze the limiting power of our tests under local alternatives. Finally, we showcase the superior size accuracy and computational efficiency of our methods as compared to some existing ones.

翻译：本文研究多元或函数型时间序列背景下的非参数推断问题，包括拟合优度检验、边缘分布变点检验及两个时间序列的独立性检验等。现有文献中的多数方法通过采用依赖于带宽的自举法或子抽样法处理这些问题，这些方法计算成本较高且/或对带宽选择敏感。为克服这些局限，我们提出一类基于核函数的新型检验方法：通过将数据嵌入再生核希尔伯特空间，并利用样本分割、投影与自标准化技术构建检验统计量。通过一种新的条件化技术，我们证明在绝对正则性与温和矩假设下，所提检验统计量具有枢轴极限零分布。同时分析了局部备择假设下检验的极限功效。最后，通过与现有方法的比较，展示了所提方法在尺度精度与计算效率方面的优越性。