# Cai–Liu–Xia maximum mean test **Status: validation-blocked; no public callable.** The oracle formula, extreme-tail, transformation, estimator-program, and invalid-input gates pass, but the practical-size Gumbel calibration does not. A reduced oracle null law passes only at a very large dimension; no honest end-to-end release scenario currently passes for the estimated-precision branches. For a precision matrix `Omega`, the oracle statistic is `n_x n_y / (n_x + n_y) * max_j ((Omega d)_j² / Omega_jj)`. The p-value uses the limiting type-I extreme-value distribution and is evaluated with `expm1` to avoid cancellation. A literal independent fixture checks the statistic, log-tail, group exchange, and diagonal feature scaling with the correspondingly transformed precision matrix. The direct paper locations are Equation (2), p. 352, for the oracle statistic; Equations (6)–(7), pp. 354–355, for the data-driven statistic and transformed variance; and Theorem 1, pp. 355–356, for `M - 2 log(p) + log(log(p))` and its type-I extreme-value limit. The CLIME program and minimum-magnitude symmetrization are displayed on pp. 353–354. These comparisons use the authors' article, not the legacy R output as a formula oracle. ## Estimated precision - `precision="clime"` solves the column-wise CLIME linear programs with SciPy, applies the paper's minimum-magnitude symmetrization, and uses `lambda = sqrt(log(p) / (n_x + n_y))`, i.e. the theoretical rate with constant one. The paper permits cross-validation of that constant; pySHT intentionally makes the deterministic rule explicit and does not silently run a data-dependent search. - `precision="adaptive-threshold"` uses the entry-specific thresholds with default `delta=2`, followed by a documented positive-definite numerical floor before inversion. - Both estimated branches assume a common covariance matrix. No unequal- covariance fallback is implied. All groups use one deterministic feature-wise anchor before per-feature scaling. Supplied precision matrices remain defined in the original column units; an explicit domain error is raised if transforming such a matrix would overflow float64. ## Null calibration audit and current limitation Under independent Gaussian coordinates and a supplied identity precision, the oracle statistic reduces exactly to the maximum of `p` independent chi-square-one variables. This gives an exact, allocation-free simulator for the implemented limiting p-value. With 20,000 null datasets per seed: | p | Seed | alpha=0.01 | alpha=0.05 | alpha=0.10 | Gate | |---:|---:|---:|---:|---:|---| | 10,000 | 20260810 | 0.00770 | 0.04650 | 0.09595 | pass | | 10,000 | 20260811 | 0.00810 | 0.04385 | 0.08920 | fail at 0.10 | | 300,000 | 20260810 | 0.00810 | 0.04905 | 0.09825 | pass | | 300,000 | 20260811 | 0.00825 | 0.04515 | 0.09170 | pass | The extreme-value approximation converges slowly. The passing `p=300,000` row uses the exact reduced oracle law; the current dense precision-matrix API cannot make that an honest end-to-end execution scenario. No null-calibration claim is therefore made for `precision="clime"` or `precision="adaptive-threshold"`. The method remains private until a practical sparse-covariance scenario passes the same gate; pySHT never substitutes a different calibration silently. ## Alternative-power status No public alternative-power result is reported. The implementation remains private as `_clx_2samp`, is absent from `pysht.mean.__all__`, and cannot pass the public release gate until an end-to-end sparse-covariance null scenario passes. A strong-alternative result would not repair that missing type-I-error evidence, so the power audit deliberately does not promote or advertise it. Primary reference: T. T. Cai, W. Liu, and Y. Xia, *Two-Sample Test of High Dimensional Means under Dependence*, JRSS B 76 (2014), 349–372, doi:10.1111/rssb.12034; author copy .