\begin{table}[p]
\centering\scriptsize\setlength{\tabcolsep}{2.6pt}
\begin{threeparttable}
\caption{Monte Carlo, exp design, $p = 1$, $N = 4,194,304$: scaled gap to $\widehat\theta_N$ (root mean square and median) and 95\% coverage by local sample size $m$.}\label{tab:mc_exp_p1}
\begin{tabular}{@{}lrrrrrrrrrrrr@{}}
\toprule
Estimator & \multicolumn{3}{c}{$m = 16$, $K = 262,144$} & \multicolumn{3}{c}{$m = 64$, $K = 65,536$} & \multicolumn{3}{c}{$m = 256$, $K = 16,384$} & \multicolumn{3}{c}{$m = 1024$, $K = 4,096$} \\
 & gap & median & cov & gap & median & cov & gap & median & cov & gap & median & cov \\
\midrule
(1,0) plain average & 136.531 & 136.513 & 0.000 & 32.502 & 32.496 & 0.000 & 8.031 & 8.030 & 0.000 & 2.003 & 1.999 & 0.468 \\
(1,1) average + 1 Newton & 9.102 & 9.100 & 0.000 & 0.516 & 0.516 & 0.930 & 0.032 & 0.031 & 0.944 & 0.002 & 0.002 & 0.952 \\
(1,2) average + 2 Newton & 0.040 & 0.040 & 0.936 & 1.30e-4 & 1.30e-4 & 0.948 & 4.85e-7 & 4.84e-7 & 0.944 & 1.88e-9 & 1.86e-9 & 0.952 \\
(2,0) half-block jackknife & 19.523 & 19.497 & 0.000 & 1.098 & 1.053 & 0.820 & 0.144 & 0.096 & 0.942 & 0.065 & 0.046 & 0.950 \\
(2,1) half-block + 1 Newton & 0.186 & 0.186 & 0.930 & 6.55e-4 & 5.41e-4 & 0.948 & 1.79e-5 & 4.52e-6 & 0.944 & 3.69e-6 & 1.06e-6 & 0.952 \\
(2,2) half-block + 2 Newton & 1.71e-5 & 1.69e-5 & -- & 2.93e-10 & 1.43e-10 & -- & 6.25e-13 & 0 & -- & 2.90e-13 & 0 & -- \\
(3,0) quarter-block & 6.557 & 6.575 & -- & 0.380 & 0.252 & -- & 0.177 & 0.123 & -- & 0.092 & 0.064 & -- \\
(3,1) quarter-block + 1 Newton & 0.022 & 0.021 & 0.936 & 1.22e-4 & 3.09e-5 & 0.948 & 2.64e-5 & 7.35e-6 & 0.944 & 7.47e-6 & 2.02e-6 & 0.952 \\
(2,1) valid workers only & -- & -- & -- & -- & -- & -- & -- & -- & -- & -- & -- & -- \\
(3,1) valid workers only & -- & -- & -- & -- & -- & -- & -- & -- & -- & -- & -- & -- \\
analytic Cox--Snell + 1 Newton & 1.31e-4 & 3.02e-5 & 0.932 & 2.44e-5 & 8.12e-6 & 0.948 & 7.50e-6 & 1.88e-6 & 0.944 & 1.59e-6 & 4.66e-7 & 0.952 \\
CSL, 1 round & 2.99e-13 & 0 & -- & 2.96e-13 & 0 & -- & 3.05e-13 & 0 & -- & 2.87e-13 & 0 & -- \\
CSL, 2 rounds & 2.88e-13 & 0 & -- & 2.99e-13 & 0 & -- & 3.22e-13 & 0 & -- & 2.86e-13 & 0 & -- \\
AEE (Lin--Xi) & 120.465 & 120.444 & -- & 31.499 & 31.496 & -- & 7.968 & 7.965 & -- & 1.999 & 1.996 & -- \\
(2,1) stale Hessian & 0.971 & 0.970 & -- & 0.016 & 0.016 & -- & 5.43e-4 & 3.75e-4 & -- & 6.40e-5 & 4.48e-5 & -- \\
centralized MLE & 0 & -- & 0.932 & 0 & -- & 0.948 & 0 & -- & 0.944 & 0 & -- & 0.952 \\
\bottomrule
\end{tabular}
\begin{tablenotes}\footnotesize
\item 500 paired replications per column. gap $=$ root mean square of $\sqrt N\,\lVert\tilde\theta - \widehat\theta_N\rVert$ over all replications (including those with failed local fits); median $=$ its median; cov $=$ empirical coverage of nominal 95\% Wald intervals (observed information at the estimate; `--' when not computed). Monte Carlo standard errors of the gaps and RMSEs are in \texttt{results/processed/summary.csv}; coverage standard error $\approx 0.010$ or less.
\item Fraction of workers with all local fits valid (orders 2 / 3): $m=16$: 1.000 / 1.000; $m=32$: 1.000 / 1.000; $m=64$: 1.000 / 1.000; $m=128$: 1.000 / 1.000; $m=256$: 1.000 / 1.000; $m=512$: 1.000 / 1.000; $m=1024$: 1.000 / 1.000. Rows `valid workers only' drop the workers with a failed sub-block fit from the pilot (weights renormalized) and coincide with the corresponding rows when no fit failed.
\end{tablenotes}
\end{threeparttable}
\end{table}