Cycle 94, stuff 1 (member-2): is SW's skew advantage general? SW vs plain AD vs plain S2, exact 5% size, 20000 paired sims per n, seed 94201

One default_rng(94201) stream consumed in order n=25, 50, 100; per n the laws normal, chi2(2), chi2(10), lognormal(0, 0.5), beta(2, 5). All three tests are scored on the same draws in every cell. Runner: stuff/cycle94_skew_general.py (uv run --with numpy --with scipy python stuff/cycle94_skew_general.py), wall clock 25 s. No cell edited by hand; the tables below are the script's output.

Definitions

z = sample-standardized order statistics (sample mean, n-1 sd, sorted ascending), i = 1..n. Plain AD term t_i = -((2i-1)/n) [ln Phi(z_(i)) + ln(1 - Phi(z_(n+1-i)))]. A2 = sum t_i - n (plain AD, reject large); S2 = sum log(1 + 2 t_i)/2 (plain S2, c = 2, reject large); W = Shapiro-Wilk (scipy.stats.shapiro, reject small). Each 5% critical value is the empirical 95% (W: 5%) quantile of that statistic's 20000 null values, so every power row is at matched exact size. MC SE of a null rate at 0.05 is 0.0015.

Seed and exact-size 5% critical values (rule 2, before any power table): seed 94201, 20000 N(0,1) draws per n

Power at exact 5% size (rule 1), 20000 paired draws per cell

Gap = SW minus other; paired SE = sd of the per-draw rejection-indicator difference / sqrt(20000). Verdict: beats if gap > 2 SE, loses if gap < -2 SE, else tie (within two paired SE).

Reading the table

1. The advantage is general, not chi-square(5)-specific. SW beats both plain AD and plain S2 by more than two paired SE in 22 of the 24 comparisons, across strong skew (chi2(2)), mild skew (chi2(10)), log-normal and the bounded beta(2, 5).
2. The gap over S2 grows with n on the mild-skew laws. SW minus S2 goes from +0.035 to +0.150 to +0.232 on chi2(10) and from +0.078 to +0.290 to +0.577 on beta(2, 5). S2's log compression of the AD terms damps exactly the moderate one-sided deviations that skew produces, so its skew power stalls while SW's climbs.
3. The only non-beats cells are saturation ties. At chi2(2), n=100 all three tests reject on essentially every draw (SW 1.0000, AD 1.0000, S2 0.9999), so the gaps are 0.0000 and 0.0001 with paired SE 0.0000 and 0.0001. SW-AD is identical on all 20000 draws (SE undefined, reported as 0.0000); SW-S2 is +1.4 SE. These are ceiling effects, not evidence of parity in finite power.
4. SW never loses. No cell at any n has SW below AD or S2 by more than two paired SE.

Answer to the direction (seed 94201)

SW does not win every skew cell at every n: it wins 22 of 24 and ties 2, both at chi-square(2), n=100, where SW minus AD is +0.0000 (0 paired SE, all draws reject under both) and SW minus S2 is +0.0001 (+1.4 paired SE) because every test is at the power ceiling; SW loses nowhere.

Cycle 94, stuff 3 (member-1): third-seed replication of SW vs plain AD vs plain S2 on skew, seed 94105

Same runner as stuff 1 (stuff/cycle94_skew_general.py) with only SEED = 94201 changed to SEED = 94105, run from a copy outside the repo with uv run --with numpy --with scipy python -I; the committed script is unchanged. 20000 paired draws per cell, one default_rng(94105) stream consumed in order n=25, 50, 100, laws normal, chi2(2), chi2(10), lognormal(0, 0.5), beta(2, 5). Definitions, exact-size calibration (empirical null quantile per statistic) and verdict rule (beats if gap > 2 paired SE, loses if gap < -2 paired SE, else tie) are unchanged. Wall clock 25 s. Tables are the script's output verbatim.

Seed and exact-size 5% critical values (rule 2): seed 94105, 20000 N(0,1) draws per n

All three null rates are exactly 0.0500 by construction. W and S2 critical values agree with seeds 94201 and 94303 to within 0.0007 and 0.007. The A2 critical value at n=100 is 0.7430 against 0.7605 for seed 94303, a 0.0175 shift: the A2 null upper tail is long, so its empirical 95th percentile is the least stable of the three. It does not change any power verdict below.

Power at exact 5% size (rule 1), 20000 paired draws per cell

Replication against seeds 94201 and 94303

1. 23 of 24 verdicts match both earlier seeds. The one difference is SW minus S2 at chi2(2), n=100: here +0.0003 with paired SE 0.0001 (6 discordant draws out of 20000, SW rejecting where S2 does not, none the other way), which the rule calls "beats" at 3 SE; seeds 94201 and 94303 had +0.0001 (0.0001) and called it a tie. This is a saturation cell where SW rejects all 20000 draws and S2 rejects 19994, so the verdict flips on a handful of draws. It is not evidence of a real gap and should be read as a tie in substance.
2. SW minus AD at chi2(2), n=100 is a tie again, as in both earlier seeds: both reject every draw, gap and SE are both zero.
3. Largest power shift between this seed and either earlier seed is 0.0181 (SW, beta(2, 5), n=50: 0.4869 here against 0.5050 for seed 94201), about 3.9 SE of an independent-seed difference (sqrt(2) x 0.0033). Next largest is 0.0172 (AD, beta(2, 5), n=100, against seed 94303). Both are in beta(2, 5), the law with the lowest SW power at n=25, and neither moves a verdict: the smallest non-saturated SW gap in this seed is +0.0027 (0.0004) at chi2(2), n=50, 6.8 SE.
4. SW never loses. No cell at any n has SW below AD or S2 by more than two paired SE; no cell has SW below either test by any amount.

Closing line (rule 2)

SW does not win every skew cell at every n: it beats AD and S2 in 23 of 24 cells and ties SW minus AD at chi-square(2), n=100 (paired difference +0.0000, 0.0 paired SE, both tests reject all 20000 draws); the SW minus S2 verdict at that same cell is a 3.0 SE "beats" on 6 discordant draws that the two earlier seeds scored as a tie, and SW loses nowhere. Across three seeds the answer to the direction is the same: SW's skew advantage is general, not specific to chi-square(5).

Cycle 94, stuff 2 (member-3): seed replication of SW vs plain AD vs plain S2 on skew, seed 94303

Same runner as stuff 1 (stuff/cycle94_skew_general.py) with only SEED = 94201 changed to SEED = 94303; 20000 paired draws per cell, one default_rng(94303) stream consumed in order n=25, 50, 100, laws normal, chi2(2), chi2(10), lognormal(0, 0.5), beta(2, 5). Definitions, exact-size calibration (empirical null quantile per statistic) and verdict rule (beats if gap > 2 paired SE, loses if gap < -2 paired SE, else tie) are unchanged. Wall clock 25 s. Tables are the script's output.

Seed and exact-size 5% critical values (rule 2): seed 94303, 20000 N(0,1) draws per n

All three null rates are exactly 0.0500 (MC SE 0.0015); critical values agree with seed 94201 to within 0.0005 (W), 0.010 (A2) and 0.005 (S2).

Power at exact 5% size (rule 1), 20000 paired draws per cell

Replication against seed 94201

Every one of the 24 verdicts matches stuff 1. Largest power shift between seeds is 0.0118 (S2, chi2(10), n=100), about 2.5 SE of an independent-seed difference (sqrt(2) x 0.0033); the smallest SW gap that is a "beats" in either seed is SW minus AD at chi2(2), n=50 (+0.0028 here, +0.0030 there, about 7 paired SE both times). The paired SE of zero at chi2(2), n=100 for SW minus AD is because all 20000 draws reject under both tests.

Answer to the direction (seed 94303)

SW does not win every skew cell at every n: it wins 22 of 24 and ties 2, both at chi-square(2), n=100, where SW minus AD is +0.0000 (0 paired SE; the difference is identically zero, all draws reject) and SW minus S2 is +0.0001 (+1.4 paired SE) at the power ceiling; SW loses nowhere, same as seed 94201.