In this paper, we consider the fundamental problem of testing for monotone trend in a time series. While the term "trend" is commonly used and has an intuitive meaning, it is first crucial to specify its exact meaning in a hypothesis testing context. A commonly used well-known test is the Mann-Kendall test, which we show does not offer Type 1 error control even in large samples. On the other hand, by an appropriate studentization of the Mann-Kendall statistic, we construct permutation tests that offer asymptotic error control quite generally, but retain the exactness property of permutation tests for i.i.d. observations. We also introduce "local" Mann-Kendall statistics as a means of testing for local rather than global trend in a time series. Similar properties of permutation tests are obtained for these tests as well.

翻译：本文研究时间序列中单调趋势检验这一基本问题。虽然"趋势"一词被广泛使用且具有直观含义，但首要关键在于明确其在假设检验语境中的确切定义。常用的著名检验方法是Mann-Kendall检验，但研究表明即使在大样本条件下，该方法也无法控制第一类错误率。另一方面，通过对Mann-Kendall统计量进行适当的标准化处理，我们构建了能在大样本下普遍控制渐近错误率、同时保留独立同分布观测置换检验精确性的置换检验方法。此外，我们引入"局部"Mann-Kendall统计量，用于检验时间序列中的局部趋势而非全局趋势。这些检验方法同样具有与置换检验相似的性质。