Watson \(U^2\) circular-uniformity test

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

Statistic and scope

Let \(U_{(i)}\) be the ordered angles divided by \(2\pi\), and define

\[ d_i=U_{(i)}-\frac{2i-1}{2n}. \]

The rotation-invariant statistic is

\[ U^2=\sum_{i=1}^n(d_i-\bar d)^2+\frac1{12n}. \]

Large values reject circular uniformity against omnibus nonuniform alternatives. Unlike Rayleigh, the reported alternative is not restricted to the first harmonic. The primary source is Watson (1961).

Angles are wrapped by an explicit positive finite period. Literal ordered-CDF fixtures plus row, rotation, reflection, wrapping, and degree/radian transformations verify the statistic. Monte Carlo replicates are iid circular-uniform samples with no nuisance fit; the public stream consumes \(nB\) uniform variates after the observed statistic. Ties are upper-tail, \(p=(b+1)/(B+1)\), and uncertainty fields are returned.

Complexity

Sorting the \(n\) circular scores makes each statistic \(O(n\log n)\) time and \(O(n)\) storage. A direct \(B\)-draw finite-null calibration is therefore \(O(Bn\log n)\) time with peak storage independent of \(B\).

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	199 (0.00995)	990 (0.04950)	2,042 (0.10210)
20260933	20260919	210 (0.01050)	1,000 (0.05000)	2,003 (0.10015)

Every entry passes the release tolerance. Since each stream shares one independent 99,999-statistic bank, the tolerance is \(\max\{0.005,4\sqrt{a(1-a)(1/20000+1/99999)}\}\) at nominal level \(a\) and includes reference-quantile as well as dataset uncertainty. Seed 20260928 generated 2,000 size-20 antipodal alternatives, with ten von Mises observations around zero and ten around \(\pi\), both at concentration 8. Against a 19,999-statistic bank from seed 20260927, 1,051/2,000 rejected at 0.05. The deliberately first-moment-free fixture distinguishes Watson’s omnibus role from Rayleigh; it does not state a minimum power guarantee.

The table is an empirical-bank audit, not 20,000 complete public calls. The public generator sequence and corrected p-value are independently replayed by focused fixtures.