Permutation Testing for Monotone “Trend”

ABSTRACT In this article, 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.