Modified Hermans–Rasson circular-uniformity test

Public verdict: validated and exposed as pysht.circular.hermans_rasson. This is a pySHT-native method with no SHT 0.1.9 crosswalk entry.

Primary kernel

For wrapped radians \(\theta_1,\ldots,\theta_n\), the modified statistic is

\[ T_{HR}=\frac1n\sum_{i=1}^n\sum_{j=1}^n \left[ \left|\,|\theta_i-\theta_j|-\pi\right|-\frac\pi2 -2.895\left\{|\sin(\theta_i-\theta_j)|-\frac2\pi\right\} \right]. \]

The sign and coefficient 2.895 are preserved exactly; large values reject. An explicit double-loop oracle checks the complete kernel, including diagonal terms. This is an omnibus Sobolev statistic designed for sensitivity to both unimodal and multimodal departures. The primary source is Hermans and Rasson (1985).

The angle normalization and invariance contract is the same as Watson’s: positive finite period, wrapping, rotation, reflection, row order, and unit conversion. Each Monte Carlo replicate is an iid circular-uniform sample with no fitted nuisance parameters. The isolated public generator consumes \(nB\) uniform variates after the observed statistic. Upper-tail ties count and \(p=(b+1)/(B+1)\) with reported MC uncertainty.

Complexity

The complete pairwise kernel costs \(O(n^2)\) time and \(O(n^2)\) storage per statistic. A 9,999-draw performance test guards the resulting \(O(Bn^2)\) calibration time; replicates are released immediately, so storage does not grow with \(B\).

Release evidence

At \(n=20\), each independent 20,000-dataset stream was compared with its own independently seeded 99,999-statistic reference bank:

reference seed	dataset seed	0.01	0.05	0.10
20260917	20260918	209 (0.01045)	974 (0.04870)	2,008 (0.10040)
20260933	20260919	211 (0.01055)	1,006 (0.05030)	1,985 (0.09925)

All cells pass the release tolerance. Because a finite independent reference bank is shared by the observed datasets in each stream, the tolerance is \(\max\{0.005,4\sqrt{a(1-a)(1/20000+1/99999)}\}\) at nominal level \(a\) and includes both sources of variation. On the same named antipodal fixture used for Watson–2,000 size-20 samples from seed 20260928, concentration 8–the test rejected 2,000/2,000 at 0.05 against a 19,999-statistic bank from seed 20260927. This confirms the intended multimodal direction, not uniform power.

These are empirical-bank counts rather than 20,000 literal public calls; focused fixtures independently replay the complete public Monte Carlo stream.