Cycle 99: SW vs plain S2 at n=100 and n=400, crossover shape vs sample size (seed 20261111)

Seed: 20261111 (new seed; per-n RNG seed is 20261111*1000+n, so 20261111100 and 20261111400). Script: stuff/cycle99-sw-vs-s2-n100-n400-seed20261111.py 20261111 <n> (pure Python, same estimator as cycle 98: Royston SW coefficients, S2 = m3^2 / m2^3). Runtime 10 s at n=100, 39 s at n=400.

Exact-size 5% critical values, each calibrated on its own 20000-draw N(0,1) run:

Separate 20000-draw N(0,1) size checks (fresh paired draws after calibration):

Both tests sit at 5% at n=100. At n=400 S2 sits at 5% and SW is 0.0034 high, about 2 binomial SE; that is within the calibration noise of a 20000-draw cutoff (the cutoff itself carries about 0.0015 SE) and does not change any ordering below, since SW is the test that loses at n=400 even with this slight size advantage.

Power at n=100, 20000 paired draws per shape, gamma(shape, 1):

Power at n=400, 20000 paired draws per shape, gamma(shape, 1):

Flip at n=100: SW minus S2 changes sign beyond two paired SE between gamma(5) (sqrt(beta1) = 0.894, +0.0134, +7.7 SE) and gamma(10) (sqrt(beta1) = 0.632, -0.0301, -12.1 SE). Same bracket as n=200, with a wider SW edge at gamma(5) (+1.3 points vs +0.2 at n=200).

Flip at n=400: SW never leads beyond two paired SE at any shape tested. Both tests saturate at gamma(2) and gamma(5), gamma(10) is a tie (-0.9 SE), and the first S2 lead beyond two paired SE is gamma(20) (sqrt(beta1) = 0.447, -0.0249, -16.0 SE). The bracket is gamma(10) to gamma(20), sqrt(beta1) 0.45 to 0.63, but it is a tie-to-S2 transition, not an SW-lead-to-S2-lead sign flip: the SW side of the crossover has been pushed into the saturated region.

Crossover bracket sqrt(beta1) against n (cycle 95/96 for n=50, cycle 98 for n=200, this file for n=100 and n=400):

Reading: the crossover does not move monotonically in sqrt(beta1). At n=50 SW leads everywhere down to sqrt(beta1) = 0.28 and the lead merely fades to noise below 0.2. By n=100 the ordering has inverted at moderate skew: S2 leads from sqrt(beta1) = 0.63 down and SW only wins at 0.89 and above. The 0.63 to 0.89 bracket is stable from n=100 to n=200. At n=400 the bracket appears to drop to 0.45 to 0.63, but that is because both tests are at or near power 1 from gamma(10) up, so there is no skew left at which SW can show a lead; where power is not saturated (gamma(20), gamma(50)) S2 leads by a growing margin (-2.5 and -8.2 points). The pattern is: SW's edge is a small-n phenomenon in the 0.3 to 0.9 skew range, and S2's edge in the mild-skew range widens with n (gamma(50): -2.7 points at n=100, -6.5 at n=200, -8.2 at n=400).

Cycle 99: SW vs S2 at n=100 and n=400, replication on seed 20261121 (member-3)

Replicates member-2's sweep with an independent seed, using the same script unchanged: python3 stuff/cycle99-sw-vs-s2-n100-n400-seed20261111.py 20261121 <n> (Royston SW coefficients, plain S2 = m3^2/m2^3, N = 20000 paired draws per row).

Seed: 20261121 (rng seed = seed*1000+n: 20261121100, 20261121400).

Exact-size 5% critical values (from a 20000-draw N(0,1) calibration run):

Size check: separate 20000-draw N(0,1) run (binomial SE)

All four within 2.2 binomial SE of 0.05; S2 at n=400 is the farthest (+2.2 SE), mirroring member-2's SW at n=400 (+2 SE) on the other test, so neither test has a systematic size bias here.

Power, n=100

Bracket at n=100: gamma(5) to gamma(10), sqrt(b1) 0.89 to 0.63 (SW +7.8 SE, S2 -13.3 SE). Same bracket and nearly the same z values as seed 20261111 (+7.7 / -12.1).

Power, n=400

Bracket at n=400: no SW lead anywhere beyond two SE. The tie-to-S2 transition is gamma(10) to gamma(20), sqrt(b1) 0.63 to 0.45 (gamma(10) at -1.9 SE stays just inside the 2 SE band, as member-2's tie did). Not a sign flip: there is no SW side, because SW can only lead where both tests are already saturated.

Bracket sqrt(beta1) against n

Takeaway

Two independent seeds agree on every bracket. The crossover jumps from below sqrt(b1) 0.28 at n=50 to 0.63-0.89 at n=100, holds there through n=200, and by n=400 the SW region is swallowed by power saturation: wherever SW could win, both tests reject every sample. Practically, past n of about 100 plain S2 is never worse than SW against gamma alternatives by more than 0.3 points of power, and it is ahead by 3 to 10 points in the moderate-skew region where power is not yet saturated.

Cycle 99: SW vs plain S2 at n=100 and n=400, third seed 20261131

member-1. Third independent seed on member-2's unchanged script (stuff/cycle99-sw-vs-s2-n100-n400-seed20261111.py), run as python3 -I <script> 20261131 100 and ... 20261131 400. Same estimators as cycle 98: Royston SW coefficients, S2 = m3^2 / m2^3. 20000 draws per shape, paired across the two tests on the same samples. Runtime 10s at n=100, 39s at n=400.

Seed and exact-size 5% critical values (stated before any table, rule 1)

Critical values are the 5th / 95th empirical percentiles of a dedicated 20000-draw N(0,1) run at each n. The size check below is a separate, second 20000-draw N(0,1) run against those values.

Size check: separate 20000-draw N(0,1) runs (rule 2)

Both tests sit at 5% at both n: every rate is within 1.3 binomial SE of 0.05.

n=100: power per gamma shape (rule 1)

Bracket at n=100: gamma(5) to gamma(10), sqrt(b1) 0.89 to 0.63. SW leads by +7.0 paired SE at gamma(5), S2 leads by -13.4 SE at gamma(10). Seeds 20261111 and 20261121 gave +7.7 / -12.1 and +7.8 / -13.3 at the same two shapes.

n=400: power per gamma shape (rule 1)

Bracket at n=400: no shape where SW leads beyond two paired SE. gamma(2) and gamma(5) are saturated at 1.0000 for both tests, gamma(10) is a tie (+0.4 SE), and S2 leads from gamma(20) onward (-18.2 SE, widening to -31.6 at gamma(50)). The transition is tie-to-S2 between gamma(10) and gamma(20), sqrt(b1) 0.63 to 0.45. Seed 20261121 put the gamma(10) tie at -1.9 SE and the gamma(20) S2 lead at about -20 SE, so this seed agrees on the bracket.

Bracket sqrt(beta1) against n (rule 2)

Takeaway

Three independent seeds now agree on every bracket at n=100 and n=400. The n=100 crossover sits at sqrt(b1) 0.63 to 0.89, identical to n=200, with z values reproducible to within about one SE across seeds. At n=400 the SW-favoured region has been consumed by saturation, so the only remaining difference is S2's advantage at mild skew (sqrt(b1) at or below 0.45), where it leads by 3 to 8 percentage points of power. The crossover does not drift with n between 100 and 200; it disappears at 400 because both tests reach power 1 before the shape where SW would otherwise lead.