Dempster non-exact mean tests

Status: formula, independent fixture, invariance, boundary, and advertised Gaussian null-calibration gates pass.

The one- and two-sample functions implement Dempster’s Euclidean mean-square ratio. For within-group degrees of freedom nu, the effective rank is estimated from tr(Sigma) and tr(Sigma²) and the statistic is calibrated by F(floor(r), floor(nu r)). This calibration assumes multivariate normality; it does not require an invertible sample covariance.

Audit findings

• dempster_1samp agrees with the trace formula in the pinned SHT source and an independent NumPy/SciPy implementation.

• The legacy mean2.1958Dempster computed a positive F-type ratio but passed it to a standard-normal upper tail. dempster_2samp repairs that mismatch and uses Dempster’s approximate F law.

• Both routines are invariant to a common nonzero scale; the two-sample routine is invariant to exchanging groups.

• Null centering and shared two-sample anchoring happen before numerical scaling. Tests at common locations up to 1e14 reproduce the attainable result from the translated float64 observations.

• Singular or high-dimensional covariance estimates are allowed, but the estimated trace and squared-trace correction must be positive.

Null calibration gate

The advertised scenario is independent Gaussian data with identity covariance, p=3000, and 50 covariance degrees of freedom: n=51 for the one-sample test and n_x=n_y=26 for the two-sample test. Under this null the two designs have the same independent chi-square mean component and Wishart covariance component. We sampled those sufficient statistics directly and cross-checked their formulas against the full-data fixture.

Seed	alpha=0.01	alpha=0.05	alpha=0.10
20260810	0.00885	0.04660	0.09875
20260811	0.01015	0.05025	0.09890

Each row uses 20,000 null datasets. Both one- and two-sample routines have these rates and satisfy the preregistered scale-aware release tolerance at all three levels. This gate advertises the stated Gaussian identity regime; it is not a blanket finite-sample guarantee for arbitrary spectra.

Targeted alternative-power gate

The public functions were also exercised end to end under dense Gaussian mean shifts with p=40 > n=30. For dempster_1samp, child-0 PCG64 draws 30 rows from \(N_{40}(0.35\mathbf 1,I)\) and the public call is mean.dempster_1samp(x). For dempster_2samp, it draws 30 rows from \(N_{40}(0.45\mathbf 1,I)\) and then 30 rows from \(N_{40}(0,I)\) before mean.dempster_2samp(x, y). Each named 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
dempster_1samp	2026090306	1,000	1,000 / 1.000	1,000 / 1.000	1,000 / 1.000
dempster_2samp	2026090307	1,000	1,000 / 1.000	1,000 / 1.000	1,000 / 1.000

These are strong-alternative detection checks, not comparative power claims. They are reproduced by python -m tools.mean_power_audits 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 references: A. P. Dempster, A High Dimensional Two Sample Significance Test (1958), and A Significance Test for the Separation of Two Highly Multivariate Small Samples (1960).