Exact distribution of the Mann–Kendall trend test statistic for persistent data

SummaryThe distribution-free Mann–Kendall test is widely used for the assessment of significance of trends in many hydrologic and climatic time series. Previous studies have suggested both exact and approximate formulas for the calculation of the variance of the test statistic when the data are serially correlated. This paper outlines a procedure for the calculation of the exact distribution of the Mann–Kendall trend test statistic for persistent data with an arbitrary correlation structure. The particular cases of the AR(1) (first order autoregressive) model and the Fractional Gaussian Noise (FGN) model are presented for sample sizes between 3 and 9. While it has been previously shown that the Normal distribution gives a reasonable approximation to the exact distribution for large values of sample size n, a more accurate approximation based on the Beta distribution is proposed for moderate values of n. The application of the test to small samples is illustrated by testing the significance of recent trends starting in 1990 in 58 world river flow time series. The results confirm the effect of scaling in small samples and the benefits of using the Beta distribution as an approximation.