Bai–Saranadasa trace mean tests

Status: formula, independent fixture, transformation, and boundary gates pass in the advertised high-dimensional asymptotic regime.

The statistic centers the squared Euclidean mean difference by tr(S) and uses the unbiased Wishart correction for tr(Sigma²). The two-sample version assumes a common covariance matrix. Its upper-tail normal calibration is asymptotic and is intended for increasing sample size and dimension.

Audit findings

• Literal implementations independently reproduce both statistics and p-values.

• The legacy one-sample denominator cancelled an n + 1 term where the squared-covariance correction requires n + 2. pySHT uses 2 n (n + 1) / ((n - 1)(n + 2)).

• Common scaling and orthogonal feature transformations leave the result unchanged; exchanging the two samples leaves bs_2samp unchanged.

• Null centering and shared two-sample anchoring precede numerical scaling; large-location regressions include 1e8 and 1e14.

• A nonpositive trace-variance estimate is rejected rather than clipped.

Null calibration gate

For independent Gaussian identity-covariance data, we used p=3000 and 50 covariance degrees of freedom (n=51 one-sample; n_x=n_y=26 two-sample). The normal mean component and Wishart covariance component are independent, so the audit samples those sufficient statistics directly; fixed full-data fixtures verify the same implementation formula.

Seed	alpha=0.01	alpha=0.05	alpha=0.10
20260810	0.01025	0.04840	0.09975
20260811	0.01120	0.05165	0.10040

Each row represents 20,000 null datasets. The one- and two-sample designs share these rates and pass the release tolerance at every level. Smaller dimensions can retain visible chi-square skewness, so this is a deliberately specific advertised asymptotic regime rather than a small-sample claim.

Targeted alternative-power gate

The end-to-end alternative audit retains the high-dimensional design p=40 > n=30. For bs_1samp, child-0 PCG64 draws 30 rows from \(N_{40}(0.18\mathbf 1,I)\) and calls mean.bs_1samp(x). For bs_2samp, it draws 30 rows from \(N_{40}(0.25\mathbf 1,I)\) and then 30 rows from \(N_{40}(0,I)\) before mean.bs_2samp(x, y). Each named integer seed starts a SeedSequence; child 0 is one persistent data stream and child 1 is a separate persistent PCG64 auxiliary stream, unused here. Streams advance in replication order and reset between methods. Rejection means pvalue < alpha.

Public function	Seed	Replications	alpha=0.01 count/rate	alpha=0.05 count/rate	alpha=0.10 count/rate
bs_1samp	2026090308	1,000	878 / 0.878	946 / 0.946	968 / 0.968
bs_2samp	2026090309	1,000	862 / 0.862	948 / 0.948	979 / 0.979

This strong-alternative gate shows directional sensitivity in the advertised high-dimensional path; it is not a ranking against other tests. Run python -m tools.mean_power_audits to reproduce it. The recorded engine was Python 3.12.13 with NumPy 2.5.1 and SciPy 1.18.0; the worst-case binomial standard error for 1,000 outer replications is 0.0159.

Primary reference: Z. Bai and H. Saranadasa, Effect of High Dimension: by an Example of a Two Sample Problem, Statistica Sinica 6 (1996), 311–329.