8.5 A Warning Not Found in the Textbooks
Applied exactly as written above, this correction is anti-conservative — it rejects \(H_0\) far more often than it should.
The reason is in step (iii). The threshold \(1.96/\sqrt {n}\) admits any autocorrelation that looks significant, and on a series with no correlation at all roughly one lag in twenty will clear it by chance. Those spurious values are as likely to be negative as positive, and a negative \(\rho _\ell \) shrinks the inflation factor. A factor below \(1\) deflates \(\operatorname {var}(S)\), which inflates \(Z\), which manufactures trends.
This is not a small effect. On \(4{,}000\) simulated white-noise series of length \(45\) — no trend whatever — the correction as published rejected at \(\mathbf {9.8\%}\) against a nominal \(5\%\). About a third of the computed inflation factors fell below \(1\), some close to zero.
The remedy is to constrain the factor so that it can only ever inflate the variance: \[\dfrac {n}{n^{*}} \longleftarrow \max \left (1,\ \dfrac {n}{n^{*}}\right ).\]
With that single clip the same simulation rejects at \(4.7\%\), against \(5.0\%\) for uncorrected Mann–Kendall. The justification is not merely empirical: the correction exists to account for information lost to serial dependence, and dependence cannot create information. A correction that reduces the variance is answering a question nobody asked.
Note 8.1. The lesson generalises beyond this test. A correction derived under an assumption — here, that the estimated autocorrelations are real — can do more damage than the problem it corrects when that assumption fails at random. Simulating the null is the only way to find out, and it costs a few lines of code. Any correction you apply to a test statistic should be checked this way before it is trusted.
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