\begin{table}[p]
\centering\scriptsize\setlength{\tabcolsep}{2.6pt}
\begin{threeparttable}
\caption{Monte Carlo, logit p20 design, $p = 20$, $N = 1,048,576$: scaled gap to $\widehat\theta_N$ (root mean square and median) and 95\% coverage by local sample size $m$.}\label{tab:mc_logit_p20_p20}
\begin{tabular}{@{}lrrrrrrrrrrrr@{}}
\toprule
Estimator & \multicolumn{3}{c}{$m = 128$, $K = 8,192$} & \multicolumn{3}{c}{$m = 256$, $K = 4,096$} & \multicolumn{3}{c}{$m = 1024$, $K = 1,024$} & \multicolumn{3}{c}{$m = 2048$, $K = 512$} \\
 & gap & median & cov & gap & median & cov & gap & median & cov & gap & median & cov \\
\midrule
(1,0) plain average & 323.376 & 323.451 & 0.000 & 131.255 & 131.250 & 0.000 & 29.210 & 29.203 & 0.343 & 14.338 & 14.326 & 0.719 \\
(1,1) average + 1 Newton & 49.477 & 49.512 & 0.075 & 7.699 & 7.698 & 0.889 & 0.369 & 0.368 & 0.953 & 0.088 & 0.088 & 0.954 \\
(1,2) average + 2 Newton & 1.031 & 1.032 & 0.947 & 0.025 & 0.025 & 0.952 & 5.84e-5 & 5.81e-5 & 0.953 & 3.37e-6 & 3.33e-6 & 0.954 \\
(2,0) half-block jackknife & 1.65e+5 & 5.03e+4 & 0.000 & 61.069 & 61.034 & 0.033 & 3.171 & 3.126 & 0.941 & 1.668 & 1.638 & 0.947 \\
(2,1) half-block + 1 Newton & 2.26e+11 & 5.46e+7 & 0.066 & 1.564 & 1.560 & 0.947 & 0.004 & 0.004 & 0.953 & 9.12e-4 & 7.57e-4 & 0.954 \\
(2,2) half-block + 2 Newton & 1.29e+131 & 2.42e+16 & -- & 0.001 & 0.001 & -- & 7.91e-9 & 5.18e-9 & -- & 5.12e-10 & 2.39e-10 & -- \\
(3,0) quarter-block & 3.25e+5 & 8.69e+4 & -- & 4.40e+4 & 1.67e+4 & -- & 3.404 & 3.342 & -- & 2.272 & 2.253 & -- \\
(3,1) quarter-block + 1 Newton & 1.84e+12 & 2.89e+8 & 0.899 & 4.14e+9 & 6.92e+5 & 0.008 & 0.003 & 0.003 & 0.953 & 0.002 & 0.001 & 0.954 \\
(2,1) valid workers only & 137.435 & 137.253 & -- & 1.564 & 1.560 & -- & -- & -- & -- & -- & -- & -- \\
(3,1) valid workers only & 1.84e+12 & 2.89e+8 & -- & 4.925 & 4.877 & -- & -- & -- & -- & -- & -- & -- \\
analytic Cox--Snell + 1 Newton & 0.253 & 0.249 & 0.949 & 0.009 & 0.008 & 0.952 & 7.34e-4 & 6.60e-4 & 0.953 & 3.59e-4 & 3.16e-4 & 0.954 \\
CSL, 1 round & 112.258 & 106.215 & -- & 29.269 & 27.982 & -- & 2.937 & 2.881 & -- & 1.017 & 0.989 & -- \\
CSL, 2 rounds & 141.077 & 114.985 & -- & 18.314 & 15.377 & -- & 0.691 & 0.640 & -- & 0.163 & 0.150 & -- \\
AEE (Lin--Xi) & 230.409 & 230.338 & -- & 112.533 & 112.500 & -- & 28.049 & 28.049 & -- & 14.027 & 14.022 & -- \\
(2,1) stale Hessian & 1.63e+5 & 4.41e+4 & -- & 6.341 & 6.328 & -- & 0.066 & 0.065 & -- & 0.018 & 0.018 & -- \\
centralized MLE & 0 & -- & 0.948 & 0 & -- & 0.952 & 0 & -- & 0.953 & 0 & -- & 0.954 \\
\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): $m=128$: 0.821 / 0.000; $m=256$: 1.000 / 0.674; $m=512$: 1.000 / 1.000; $m=1024$: 1.000 / 1.000; $m=2048$: 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}