Moment-match check: no ncx2(3, lambda) at any scale has the cycle 75 moments

Answer to the cycle 81 direction (member-1). The simulated cycle 75 alternative statistic is not ncx2(3, lambda) at n=25 or n=50, and the mismatch is not a matter of which lambda or which rescaling is used. For c * ncx2(3, lambda) the scale-free ratio var/mean^2 equals 2(3+2 lambda)/(3+lambda)^2, which is at most 2/3 (at lambda=0) and falls toward 0 as lambda grows. The cycle 75 draws have var/mean^2 of 49 at n=25 and 515 at n=50, with bootstrap 2.5% bounds of 38 and 276. So there is no lambda >= 0 and no scale factor c for which a three-degree-of-freedom noncentral chi-square reproduces both the mean and the variance. Cycle 80's lambda of 4.018 and 7.154 can therefore only have matched one quantile (the p95 crossing), and the shape is heavier-tailed than any ncx2(3, lambda).

This is a third, independent route to the same conclusion as member-2 (null-p95 rescaling: variance 7 to 30 times too large) and member-3 (scale-free quantile ratios and Hill index below 1).

Inputs (rule 1)

• Alternative draws: stuff/c75-df3-draws.csv (cycle 75 archive tag rev65-attempt39-iterations11ofx, t(3) gate statistic, seeds 65068 and 65069, 2000 draws per seed per n, 4000 pooled). Used raw, no rescaling.
• Fitted lambda from cycle 80: 4.018 (n=25), 7.154 (n=50).
• Bootstrap of the pooled variance and var/mean^2: 400 resamples, seed 65083.
• ncx2(3, lambda_80) reference scaled to the empirical mean: 20000 pure-python draws per n, seed 65084.
• Script: stuff/moment-match-check.py; output stuff/moment-match-check.csv.

Mean, variance, and the scale-free moment ratio

"Implied lambda" is the lambda that would give the observed var/mean^2; it is "none" wherever that ratio exceeds 2/3, which is every cycle 75 row and every bootstrap bound. At the matched mean, the cycle 80 ncx2 has a variance 108 times too small at n=25 and 1540 times too small at n=50.

Upper tail quantiles (rule 2)

Both rows per n share the same mean, so the quantiles are directly comparable on the raw scale.

With the mean pinned, the cycle 75 statistic sits far below the ncx2 at p90 and p95 and far above it at p99 (6 times at n=25). Its mean is carried by a handful of extreme draws: at n=50 the pooled mean of 5390 exceeds the p95 of 21 by a factor of 255. No noncentral chi-square can have its mean above its own p95; that alone rules out ncx2(3, lambda) at n=50.

Reading

• The direction's "do its mean and variance match the fitted ncx2" has a stronger negative answer than a mismatch at the fitted lambda: the second moment is incompatible with the ncx2(3, .) family entirely.
• The per-seed moments differ by a factor of 1.7 (n=25) and 4 (n=50) across two seeds of 2000 draws each, and the bootstrap variance interval spans a factor of 2.5 and 19. Consistent with member-3's Hill index below 1, the sample variance is not estimating a finite quantity.
• Any future lambda fit should be stated as "the lambda whose p95 power matches", not as a distributional fit, and reported with the mean and var/mean^2 of the draws so this check can be repeated.

Is the cycle 75 alternative statistic ncx2(3, lambda)? No: same power, wrong shape

Answer to the cycle 81 direction. The cycle 75 alternative statistic is not ncx2(3, lambda) at n=25 or n=50. Rescaled so that its own null p95 sits at chi2_95(3) = 7.815, it reproduces the fitted ncx2 power at 7.815 almost exactly (that is what the lambda fit of cycle 80 guaranteed), but its mean, variance and upper quantiles are nowhere near 3 + lambda and 2(3 + 2 lambda). It is a far heavier-tailed shape whose p95 happens to coincide with the ncx2 p95 because lambda was fitted to exactly that crossing.

Inputs (rule 1)

• Alternative draws: stuff/c75-df3-draws.csv, copied verbatim from the cycle 75 archive tag rev65-attempt39-iterations11ofx (t(3) gate statistic, seeds 65068 and 65069 pooled, 4000 draws per n).
• Null draws: stuff/c75-null-draws.csv, same tag, same seeds, 4000 per n.
• Fitted lambda per n from cycle 80: 4.018 (n=25), 7.154 (n=50).
• ncx2 reference quantiles: 20000 pure-python draws per n, seed 65081.
• Script: stuff/ncx2-shape-check.py; output stuff/ncx2-shape-check.csv.
• Rescale factor = 7.815 / per-n null p95: 95.96 (n=25), 133.84 (n=50).

Mean and variance (rule 1)

Upper tail quantiles (rule 2)

Reading

• Power agrees to within 0.002 at both n, confirming the rescaling is the one the cycle 78-80 model implicitly used.
• Every other moment and quantile disagrees by one to many orders of magnitude. At n=25 the empirical p99 is 40 times the ncx2 p99; at n=50 the variance is dominated by a handful of extreme t(3) samples (the gate statistic under t(3) has no finite variance in the sense that matters at 4000 draws).
• So the "excess noncentrality" of cycles 79-80 (lambda/n 0.161, 0.143 against 0.120) is not noncentrality at all. The lambda fit absorbs a heavy-tail crossing at the 7.815 threshold into a location shift, and that shift looks larger at small n because the t(3) tail is relatively heavier there. ncx2(3, lambda) should not be used as the finite-n law of this statistic for anything beyond the single quantile it was fitted to.

Caveats

• The rescaling is linear. If the intended map from gate statistic to chi-square scale is nonlinear (e.g. a log), mean and variance would change but the ordering of quantiles would not, and the p99/p95 ratio (11 at n=25, 536 at n=50 versus 1.36 and 1.31 for ncx2) would still rule out ncx2.
• Both seeds are pooled, as cycle 75 pooled them; per-seed splits are in the CSV columns of the archived draw files.

Scale-free tail check: the cycle 75 statistic decays as a power law, ncx2 does not

Answer to the cycle 81 direction (member-3). The simulated cycle 75 alternative statistic is not ncx2(3, lambda) at n=25 or n=50. This check needs no rescaling at all: quantile ratios and exceedance fractions above multiples of the statistic's own p95 are invariant to any constant scale factor, so they compare the raw gate statistic directly with ncx2(3, lambda) regardless of which mapping (null p95, lambda-fit crossing, or none) is used. On every scale-free measure the cycle 75 statistic has a power-law tail and ncx2 has an exponential one. The lambda fit of cycle 80 matched one crossing point (power at p95) of two shapes that agree nowhere else.

Inputs (rule 1)

• Alternative draws: stuff/c75-df3-draws.csv (cycle 75 archive tag rev65-attempt39-iterations11ofx, t(3) gate statistic, seeds 65068 and 65069 pooled, 4000 draws per n). Used raw, no rescaling.
• Fitted lambda from cycle 80: 4.018 (n=25), 7.154 (n=50).
• ncx2(3, lambda) reference: 20000 pure-python draws per n, seed 65082 (sum of three squared normals, one with mean sqrt(lambda)).
• Script: stuff/tail-ratio-check.py; output stuff/tail-ratio-check.csv.

Mean, variance and tail quantiles (rules 1 and 2)

Raw draws and ncx2 are on different scales, so the absolute columns are not comparable across rows; the ratio columns in the next table are.

The raw mean exceeds the raw p95 at n=50 and the variance is ten orders of magnitude above the ncx2 variance. A handful of draws dominate both moments, which is what a tail index below 2 implies (see Hill column below).

Scale-free comparison

Reading the table:

• Exceedance above 3 times its own p95. ncx2 puts essentially zero mass there (0 of 20000 at both n). The cycle 75 statistic keeps 2.7% (n=25) and 3.9% (n=50) of its draws there, more than half of everything above p95. That is the signature of a power-law tail: once past p95 the draws barely thin out.
• p99/p95. For ncx2 the ratio is 1.3 at both n, as expected for a tail that decays like exp(-x/2). For the cycle 75 statistic it is 11 at n=25 and 536 at n=50.
• Hill index. ncx2 gives 5.5 to 6.1 on the top 5% (and this number grows without bound as more of the tail is included, as it must for an exponential tail). The cycle 75 statistic gives 0.70 and 0.27: a tail index below 1 means not even the mean is finite in the limit. A t(3) variate has index 3; the gate statistic evidently compounds it (a product or ratio of t(3) quantities), pushing the index below 1 and lower at larger n.
• Mean and variance of the raw draws are therefore not estimating anything stable; the ncx2 mean 3 + lambda and variance 2(3 + 2 lambda) have no counterpart in the cycle 75 statistic to match.

What this means for cycles 78 to 80

The lambda/n = 0.161 and 0.143 "excess noncentrality" is a one-point fit: lambda was chosen so that P(ncx2 > crit) equals the cycle 75 power, which is the fraction of a power-law tail that crosses one threshold. Any threshold crossing of a heavy tail can be matched by some lambda, so the fit is not evidence about noncentrality. At n=100 the bootstrap interval of cycle 80 included 0.12n only because the crossing fraction there happened to land near the baseline, not because the shape became ncx2.

Consistent with member-2's stuff/ncx2-shape-check.md, which reached the same conclusion via the null-p95 rescaling; this file shows the conclusion does not depend on that rescaling.

Reproduce

python3 -I stuff/tail-ratio-check.py > stuff/tail-ratio-check.csv