# On the Gaussian distribution of the Mann-Kendall tau in the case of autocorrelated data

## TL;DR

This paper shows that the Gaussian assumption for the Mann-Kendall test breaks down for finite-length autocorrelated time series, and provides practical criteria to assess when this assumption is valid.

## Contribution

The paper proves that the normalized Mann-Kendall tau converges to a non-Gaussian distribution for AR(1) and SMA processes under certain asymptotic conditions.

## Key findings

- The normalized Mann-Kendall tau converges to a bounded distribution with positive variance, not Gaussian, for AR(1) and SMA processes.
- Shapiro-Wilk tests confirm the departure from normality in finite-length time series.
- Practical criteria are established to assess the validity of the Gaussian assumption based on autocorrelation and series length.

## Abstract

Non-parametric Mann-Kendall tests for autocorrelated data rely on the assumption that the distribution of the normalized Mann-Kendall tau is Gaussian. While this assumption holds asymptotically for stationary autoregressive processes of order 1 (AR(1)) and simple moving average (SMA) processes when sampling over an increasingly long period, it often fails for finite-length time series. In such cases, the empirical distribution of the Mann-Kendall tau deviates significantly from the Gaussian distribution. To assess the validity of this assumption, we explore an alternative asymptotic framework for AR(1) and SMA processes. We prove that, along upsampling sequences, the distribution of the normalized Mann-Kendall tau does not converge to a Gaussian but instead to a bounded distribution with strictly positive variance. This asymptotic behavior suggests scaling laws which determine the conditions under which the Gaussian approximation remains valid for finite-length time series generated by stationary AR(1) and SMA processes. Using Shapiro-Wilk tests, we numerically confirm the departure from normality and establish simple, practical criteria for assessing the validity of the Gaussian assumption, which depend on both the autocorrelation structure and the series length. Finally, we illustrate these findings with examples from existing studies.