EHY rectangular-uniformity test

Public verdict: validated and exposed as pysht.uniformity.ehy. It is a pySHT-native addition with no SHT 0.1.9 crosswalk entry.

Primary equations and hypotheses

The fixed null is uniformity on the user-declared hyperrectangle. Coordinates are mapped affinely to the unit cube, where the null density is \(f_0=1\). For intrinsic dimension \(m\), let \(v_m=\pi^{m/2}/\Gamma(m/2+1)\) and let \(X_i^{(k)}\) be the \(k\)th nearest neighbor of \(X_i\). The implementation follows EHY equations (1) and (4):

\[ \xi_{n,J}^{(\alpha)}(X_i)= \sum_{k=1}^{J}\left\{v_m n \lVert X_i-X_i^{(k)}\rVert^m\right\}^{\alpha}, \qquad T_{n,J}^{(\alpha)}= \sum_{i=1}^n\xi_{n,J}^{(\alpha)}(X_i) f_0(X_i)^\alpha. \]

Thus n_neighbors=J means all first \(J\) terms are summed. It never means only the \(J\)th distance. Theorem 1 and remarks (i), (iv), and (v) require \(\alpha>0\), identify \(\alpha=1\) as distribution-free and unusable for this test, and establish lower-tail rejection for \(0<\alpha<1\) and upper-tail rejection for \(\alpha>1\). The domain is \(1\le J<n\). The paper permits \(m=1\); pySHT therefore accepts one-feature rectangles rather than inheriting the unrelated \(d\ge2\) Yang–Modarres restriction.

The primary source is Ebner, Henze, and Yukich (2018).

Numerical and calibration policy

The sum is evaluated in the log domain; coincident observations and extreme positive powers therefore have explicit zero/infinite-statistic behavior without intermediate overflow. Bounds are fixed, finite, and not estimated. Monte Carlo replicates are iid uniform matrices on the standardized cube; no nuisance parameter is refitted. A public call consumes \(nmB\) uniform variates from its isolated generator after evaluating the data. Ties enter the selected tail and \(p=(b+1)/(B+1)\).

Literal first-\(J\) loops, one-dimensional fixtures, row/feature permutations, per-coordinate affine maps, duplicates, tail direction, seeded replay, and a 9,999-draw performance test cover the implementation.

Complexity

For \(n\) observations in \(m\) dimensions, one statistic evaluation costs \(O(n^2m)\) time and \(O(n^2m)\) peak storage for the pairwise coordinate differences. A \(B\)-draw calibration therefore costs \(O(Bn^2m)\) time while processing and releasing one null replicate at a time, so storage does not grow with \(B\).

Release evidence

At \((n,m,J)=(25,2,2)\), each independent 20,000-dataset stream was compared with its own independently seeded 99,999-statistic reference bank:

\(\alpha\)	reference seed	dataset seed	0.01	0.05	0.10
0.5	20260914	20260915	207 (0.01035)	1,035 (0.05175)	1,973 (0.09865)
0.5	20260932	20260916	189 (0.00945)	973 (0.04865)	1,982 (0.09910)
2.0	20260914	20260915	178 (0.00890)	986 (0.04930)	1,991 (0.09955)
2.0	20260932	20260916	207 (0.01035)	966 (0.04830)	1,986 (0.09930)

All pass the release tolerance. Because a finite reference bank is shared within each stream, that tolerance is \(\max\{0.005,4\sqrt{a(1-a)(1/20000+1/99999)}\}\) at nominal level \(a\); the second reciprocal term accounts for reference-quantile uncertainty. Seed 20260924 generated 2,000 size-40 samples with independent coordinates from a clipped \(N(0.2,0.02^2)\) alternative. With \(\alpha=0.5\), \(J=1\), and a 19,999-statistic null bank from seed 20260923, 2,000/2,000 rejected at 0.05. This is a strong clustered-alternative check, not a universal power claim.

The table is an empirical-bank audit, not 20,000 literal public calls with 9,999 draws each. Focused tests independently replay the exact public null stream, selected tail, tie rule, and plus-one correction.