Permutation Testing for Monotone Trend

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.