Cycle 97: SW vs plain S2 at n=200, gamma(50) to gamma(1000), seed 20261101

Does the point where the Shapiro-Wilk margin over plain S2 vanishes move to milder skew as n grows from 50 to 200?

Seed: 20261101 (paired gamma sweep), 20261102 (null calibration stream), 20261103 (separate N(0,1) size check). n: 200. Paired draws per shape: 20000. Null calibration: 60000 N(0,1) draws. Exact-size 5% critical values at n=200: Shapiro-Wilk rejects when W < 0.98632. Plain S2 (squared sample skewness b1 = m3^2 / m2^3) rejects when b1 > 0.11254.

Method: pure Python, Royston (1992) AS R94 SW weights, random.gammavariate draws. Each replicate draws one gamma(shape) sample of 200 and scores it with both tests. Paired SE is sd(I_SW - I_S2)/sqrt(20000). Run time 26 s.

N(0,1) size check (separate 20000 draws)

Both tests are within 1.1 SE of 0.05. Paired difference +0.0010 (paired SE 0.0015).

Power and margin against sqrt(beta1) = 2/sqrt(shape)

Reading

At n=200 S2 beats SW at every shape in the sweep, by 6 to 23 paired SE from gamma(50) to gamma(500). The S2 lead shrinks as skew falls and is within noise by gamma(1000). At n=50 SW still led at gamma(50) by about 8 paired SE. At n=200 the same shape gives S2 a 5.8-point lead. So with more data the SW advantage does not reach milder skew. It moves the other way, to stronger skew than gamma(50) (sqrt(beta1) > 0.28). That fits SW spending its power on tail and kurtosis departures that the near-normal gammas do not have, while S2 puts all its power on skewness, the only departure they do have.

Next: to find where SW's lead comes back at n=200, sweep stronger skew (gamma(5), gamma(10), gamma(20)), not milder.

Closing line (rule 2): none. No shape from gamma(50) to gamma(1000) has SW minus S2 beyond +2 paired SE at n=200. The closest is gamma(1000) (sqrt(beta1)=0.0632, margin -0.0021, paired SE 0.0016). The boundary does not sit at milder skew than the n=50 boundary between gamma(50) and gamma(100). It sits at stronger skew than gamma(50).

Cycle 97: SW vs plain S2 at n=200, replication seed 20261102

Seed: 20261102 (critical values from seed+1, N(0,1) check from seed+2, gamma draws from seed). Exact-size 5% critical values at n=200, from 60000 N(0,1) draws: Shapiro-Wilk rejects when W < 0.98624; plain S2 (b1 = m3^2/m2^3) rejects when b1 > 0.11319. W uses Royston's approximation of the coefficients. This is the same script as the seed 20261101 item, with only the seed changed.

Size check: separate 20000-draw N(0,1) run at n=200

Paired SW − S2 = −0.0010 (paired SE 0.0015). Both tests are at 5% within one SE.

Power under gamma(shape), 20000 paired draws each

This matches seed 20261101: S2 leads at every shape, and its lead shrinks steadily as skew decreases. At n=200 SW is not ahead anywhere from sqrt(beta1) 0.28 down to 0.06. At n=50 the sign was the other way at gamma(50), where SW led. At n=200 the margin flips to S2's favour even at the strongest skew tested. So the crossover has moved to stronger skew than gamma(50), not to milder skew.

Boundary: none. No shape from 50 to 1000 has SW − S2 above two paired SE at n=200. So the boundary does not sit at milder skew than the n=50 boundary between gamma(50) and gamma(100). If it exists, it lies at stronger skew, below shape 50.

Cycle 97: SW vs plain S2 at n=200, third seed 20261103

Seed: 20261103 (critical values from seed+1, N(0,1) check from seed+2, gamma draws from seed). Exact-size 5% critical values at n=200, from 60000 N(0,1) draws: Shapiro-Wilk rejects when W < 0.98633; plain S2 (b1 = m3^2/m2^3) rejects when b1 > 0.11288. W uses Royston's approximation of the coefficients. This is the same script as the seed 20261101 and 20261102 items, with only the seed changed.

Size check: separate 20000-draw N(0,1) run at n=200

Paired SW − S2 = −0.0004 (paired SE 0.0015). Both tests are at 5% within one SE.

Power under gamma(shape), 20000 paired draws each

All three seeds agree: S2 leads at every shape from sqrt(beta1) 0.28 down to 0.06. The margins at shapes 50, 100 and 200 match the other two seeds to within one paired SE. At shapes 500 and 1000 the S2 lead is still beyond three paired SE.

Boundary: none. SW does not beat S2 by more than two paired SE at any shape from 50 to 1000 at n=200. It does not sit at milder skew than the n=50 boundary between gamma(50) and gamma(100). The crossover moves the other way: SW led at gamma(50) when n=50, but trails there at n=200. So any n=200 crossover lies below shape 50, at stronger skew.