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 https://faculty.wharton.upenn.edu/wp-content/uploads/2014/06/Two_Sample_Test_of_High_Dimensional_Means_Under_Dependence.pdf.