Srivastava–Du diagonal mean tests

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

These tests replace the full covariance inverse with marginal variance standardization and estimate the calibration variance through tr(R²), where R is the sample correlation matrix. Every tested feature must have positive variance. The two-sample form assumes a common covariance matrix.

The independent ledger recomputes the diagonal quadratic form, finite-sample centering, tr(R²) correction, and standard-normal tail without using pySHT helpers. It also checks invariance to separate positive feature rescalings and to exchanging the two groups.

Null vectors are removed before scaling and two groups use one deterministic feature-wise anchor. Regressions at common locations 1e8 and 1e14 compare against the information still representable in the translated float64 input.

Null calibration gate

The advertised high-dimensional scenario uses independent Gaussian identity-covariance data with p=100 and 50 covariance degrees of freedom: n=51 for sd_1samp and n_x=n_y=26 for sd_2samp. The audit directly samples the independent normal mean component and Wishart covariance component, including the complete sample-correlation correction.

Seed	alpha=0.01	alpha=0.05	alpha=0.10
20260810	0.01245	0.05080	0.09605
20260811	0.01260	0.05015	0.09495

Each row uses 20,000 null datasets. The one- and two-sample designs share these rates and pass the release tolerance at all three levels. This does not extend the calibration claim to arbitrary covariance spectra.

Targeted alternative-power gate

The public standardized-coordinate paths were run with p=40 > n=30 and identity covariance. For sd_1samp, child-0 PCG64 draws 30 rows from \(N_{40}(0.18\mathbf 1,I)\) before mean.sd_1samp(x). For sd_2samp, it draws 30 rows from \(N_{40}(0.25\mathbf 1,I)\) and then 30 rows from \(N_{40}(0,I)\) before mean.sd_2samp(x, y). Each named integer seed initializes a SeedSequence; child 0 is the 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
sd_1samp	2026090310	1,000	848 / 0.848	934 / 0.934	963 / 0.963
sd_2samp	2026090311	1,000	824 / 0.824	935 / 0.935	958 / 0.958

These deliberately strong alternatives test response direction, not relative efficiency. python -m tools.mean_power_audits reproduces the counts under Python 3.12.13, 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: M. S. Srivastava and M. Du, A Test for the Mean Vector with Fewer Observations than the Dimension, Journal of Multivariate Analysis 99 (2008), 386–402.