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 strong mixing 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：本文研究了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、边缘分布中变点的存在性检验以及两个时间序列的独立性检验等。现有文献中的大多数方法通过采用依赖于带宽的自举法或子抽样法来解决这些问题，这些方法计算成本较高且/或对带宽选择敏感。为克服这些局限性，我们提出了一类新的基于核的检验方法，通过将数据嵌入再生核希尔伯特空间，并利用样本分割、投影和自归一化技术构建检验统计量。通过一种新的条件化技术，我们证明了在强混合和温和矩假设下，我们的检验统计量具有枢轴极限零分布。我们还分析了在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，展示了我们方法在尺寸精度和计算效率方面的优越性。