Cycle 89: c grid on the Anderson-Darling term scale, seed 89101 (member-1)

Definition. For a sample of size n, let z_(1) <= ... <= z_(n) be the order statistics after sample standardization (mean subtracted, divided by the ddof=1 standard deviation). The Anderson-Darling term is

t_i = -((2i-1)/n) * [ln Phi(z_(i)) + ln(1 - Phi(z_(n+1-i)))], i = 1..n.

The statistic at c is sum_i t_i for c = 0 and sum_i log(1 + c t_i)/c otherwise. Critical values are the 95th percentile of the statistic under N(0,1), so every row has normal size 0.0500 by construction. Power is the rejection rate at that normal-calibrated 5% critical value. The gain over c=0 is the paired difference in rejection indicators on the same draws, and the paired SE is the standard error of that difference across sims.

Settings. Seed 89101 (new; member-2 and member-3 used other seeds). 20000 sims per n per law. c in 0, 0.25, 0.5, 1, 2, 3, 5, 10. n in 25, 50, 100. Laws: normal, t(3), t(5), 0.9 N(0,1) + 0.1 N(0,9) (contaminated normal, drawn as N(0,1) scaled by 3 with probability 0.1). Script: numpy + scipy log_ndtr, run via uv outside the repo.

Tables (one row per c, n, law)

n = 25

n = 50

n = 100

Rule 2 answers

c maximizing t(3) gain over c=0, per n:

Monotonicity over c = 0.25, 0.5, 1, 2, 3: not monotone at any n. The first turn is at n=25, c=2 (gain drops from +0.0790 at c=1 to +0.0787 at c=2). At n=50 and n=100 the first turn is at c=3. Every turn below c=5 is smaller than one paired SE, so the plateau from c=1 to c=3 is flat within Monte Carlo noise. Past c=3 the t(3) gain falls at every n through c=5 and c=10, and those drops exceed two SE.

t(3)-optimal c under t(5) and under 0.9 N(0,1) + 0.1 N(0,9): gains, never losses, at every n, each far beyond two paired SE.

Under t(5) the best c is 3 at every n, and under the contaminated normal the best c drifts upward with n (3 at n=25, 5 at n=50, 10 at n=100). But at the t(3)-optimal c the shortfall against those law-specific optima is at most 0.02 in power, while the gain over c=0 is 0.07 to 0.20.

Verdict

Fix c = 2. It is the t(3)-optimal c at n=50 and n=100 and trails c=1 at n=25 by 0.0003, far inside noise. It agrees with member-3's seed-89303 verdict. c=2 gains by more than two paired SE under every non-normal law at every n, and the direct c=2 vs c=3 gap is within one SE of the gain at every n, so the choice between 1, 2 and 3 is not resolvable at 20000 sims; c=2 sits in the middle of the plateau.

Caveat. The paired SE shown is for the gain over c=0, not for the direct difference between adjacent c values. The gaps between c=1, 2, 3 are all under 0.0013 and should be read as ties.

Cycle 89: Anderson-Darling term c grid, independent seed (member-3)

Second, independent run of the cycle 89 grid. Different seed and generator from cycle89-ad-term-c-grid.md (numpy PCG64 here, Python random there).

Definition

With z the sample-standardized order statistics (mean, sd with ddof=1) of a sample of size n,

t_i = -((2i-1)/n) * [ln Phi(z_(i)) + ln(1 - Phi(z_(n+1-i)))], i = 1..n.

Statistic: S_c = sum_i log(1 + c t_i)/c for c > 0, and S_0 = sum_i t_i. Reject when S_c exceeds its 95% quantile under N(0,1) at the same n (normal-calibrated, 5% size). Gain = power(c) - power(c=0) on the same samples; paired SE = sd(1[S_c rej] - 1[S_0 rej]) / sqrt(20000).

Seed 89303 (numpy default_rng, one stream drawn in order n, then normal, t(3), t(5), cn). Sim count: 20000 per n per law. cn = 0.9 N(0,1) + 0.1 N(0,9).

Results

Per n: which c maximizes t(3) gain, monotonicity over 0.25..3, and t(5)/cn at the t(3)-optimal c.

First n and c where the t(3) gain turns: n=25, c=3. At every n the gain falls from c=3 to c=5 to c=10, and c=10 still gains over c=0 by more than 20 SE.

The t(3)-optimal c gains under t(5) and under the contaminated normal by more than two paired SE at every n. It never loses.

c=2 and c=3 are within 0.004 of each other at every n. The difference at n=50 (0.0003) is far below one SE. That SE is for the gain over c=0, not the c=2 vs c=3 difference, so it is only a rough reference.

Verdict

Fix c = 2. It maximizes t(3) gain at n=25 and n=100, and is within 0.0003 of the maximum at n=50. It gains under t(5) and under 0.9 N(0,1) + 0.1 N(0,9) by more than two paired SE at every n.

Full table (power, gain over c=0, paired SE)

n = 25

n = 50

n = 100

Cycle 89: best c below 20 on the Anderson-Darling term scale

Definition. For a sample of size n, standardize by the sample mean and sample sd (n-1), sort to get z_(1) <= ... <= z_(n), and set the Anderson-Darling term t_i = -((2i-1)/n) * [ln Phi(z_(i)) + ln(1 - Phi(z_(n+1-i)))]. The statistic is S_c = sum_i log(1 + c t_i)/c for c > 0 and S_0 = sum_i t_i. Rejection is at the 5% size calibrated on the normal law: the critical value for each (c, n) is the 95th percentile of S_c over the 20000 normal sims at that n, and the same 20000 sims serve as the normal row. Power is the fraction of sims exceeding that critical value. Gain over c=0 is the mean of the paired rejection difference 1[S_c rejects] - 1[S_0 rejects] on the same samples, with SE the sd of that difference over sqrt(20000).

Seed 890421 (job seed = 890421*1000 + 10*index(n) + index(law), Python random.Random). Sim count: 20000 per (n, law), so 20000 per n per law. Laws: normal, t(3), t(5), cn = 0.9 N(0,1) + 0.1 N(0,9). Grid c in 0, 0.25, 0.5, 1, 2, 3, 5, 10; n in 25, 50, 100. Pure Python 3, math.erfc for Phi.

Rule 2 items

• n=25: the c maximizing t(3) gain over c=0 is c=2 (gain +0.0780, SE 0.0022). t(3) gain over c in 0.25, 0.5, 1, 2, 3 is not monotone: 0.25: +0.0427, 0.5: +0.0611, 1: +0.0718, 2: +0.0780, 3: +0.0741. At c=2, under t(5) the gain gains by more than two paired SE (+0.0600, SE 0.0020); under 0.9 N(0,1) + 0.1 N(0,9) the gain gains by more than two paired SE (+0.0814, SE 0.0022).
• n=50: the c maximizing t(3) gain over c=0 is c=2 (gain +0.1182, SE 0.0024). t(3) gain over c in 0.25, 0.5, 1, 2, 3 is not monotone: 0.25: +0.0717, 0.5: +0.0969, 1: +0.1131, 2: +0.1182, 3: +0.1168. At c=2, under t(5) the gain gains by more than two paired SE (+0.1399, SE 0.0026); under 0.9 N(0,1) + 0.1 N(0,9) the gain gains by more than two paired SE (+0.1499, SE 0.0026).
• n=100: the c maximizing t(3) gain over c=0 is c=2 (gain +0.0791, SE 0.0019). t(3) gain over c in 0.25, 0.5, 1, 2, 3 is not monotone: 0.25: +0.0567, 0.5: +0.0675, 1: +0.0751, 2: +0.0791, 3: +0.0786. At c=2, under t(5) the gain gains by more than two paired SE (+0.1999, SE 0.0029); under 0.9 N(0,1) + 0.1 N(0,9) the gain gains by more than two paired SE (+0.1500, SE 0.0026).

Monotonicity: the t(3) gain is not monotone in c over 0.25, 0.5, 1, 2, 3; the first turn is at n=25, c=3 (gain drops from the previous c).

Verdict: fix c=2. It maximizes t(3) gain at every n and never loses by more than two paired SE under t(5) or the contaminated normal.