# P13091: Finite-sample distribution of Mann–Kendall rank statistics for stationary Gaussian sequences

- ID: `P13091`
- Reference: `finite-sample-mann-kendall-stationary-gaussian`
- Page: https://theoremdb.org/statements/P13091
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/finite-sample-mann-kendall-stationary-gaussian?representation=markdown
- Record maturity: Reviewed problem

## The problem

Let \((X_t)_{t\in\mathbb Z}\) be a real-valued, continuous, zero-mean, unit-variance, strictly stationary Gaussian process with autocorrelation function \(r(k)=\operatorname{Cov}(X_0,X_k)\), where \(r(0)=1\) and \(r\) is positive semidefinite. For \(n\ge3\), define \(S_n=\sum_{1\le i<j\le n}\operatorname{sgn}(X_j-X_i)\) and \(\tau_n=2S_n/[n(n-1)]\). Under the null hypothesis that there is no deterministic monotone trend, determine the finite-sample distribution \(F_{n,r}(s)=\Pr_r(S_n\le s)\). The target is a general finite-sample distributional theory, not merely the formal expression as a sum of Gaussian ordering probabilities. A successful positive resolution must provide a mathematically justified representation or algorithm for the CDF and/or quantiles over a clearly specified nontrivial class of stationary Gaussian covariance functions, with either exact evaluation or certified numerical error, and with provable computational complexity asymptotically smaller than explicit enumeration of all \(n!\) orderings. The solution must recover the iid Mann–Kendall distribution when \(r(k)=0\) for \(k\ne0\); recover known exact finite-n dependent cases, including published small-n Gaussian AR(1) results; apply to a covariance class strictly broader than a single AR(1) family; avoid explicit enumeration of all \(n!\) orderings; provide rigorous error bounds if numerical rather than exact evaluation is used; give computable tail probabilities or critical values; characterize, on at least a nontrivial covariance class, the finite-n discrepancy between the exact law and commonly used normal/variance-corrected approximations; and establish the precise conditions under which the proposed representation remains valid, including singular or near-singular covariance limits where relevant. A negative resolution is also admissible: prove, under an explicitly stated computational model and for an explicitly stated covariance class, that no subfactorial algorithm with the claimed accuracy guarantees can exist, together with a mathematically sharp characterization of the obstruction. A mere observation that the naive method has n! terms does not constitute such a resolution.

## Status

**Resolution status:** Open.

No proof or formalization has been recorded.

## Research packet

### Working on this

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## References

No external mathematical reference has been recorded for this problem.