\begin{table}[p]
\centering\scriptsize
\begin{threeparttable}
\caption{Monte Carlo, logit design, $p = 8$, $N = 1,048,576$: scaled gap to $\widehat\theta_N$, scaled RMSE and 95\% coverage by local sample size $m$.}\label{tab:mc_logit_p8}
\begin{tabular}{@{}lrrrrrrrrr@{}}
\toprule
Estimator & \multicolumn{3}{c}{$m = 256$, $K = 4,096$} & \multicolumn{3}{c}{$m = 512$, $K = 2,048$} & \multicolumn{3}{c}{$m = 1024$, $K = 1,024$} \\
 & gap & RMSE & cov & gap & RMSE & cov & gap & RMSE & cov \\
\midrule
(1,0) plain average & 44.320 & 44.777 & 0.000 & 21.491 & 22.386 & 0.148 & 10.589 & 12.356 & 0.633 \\
(1,1) average + 1 Newton & 0.835 & 6.432 & 0.950 & 0.195 & 6.304 & 0.952 & 0.047 & 6.342 & 0.953 \\
(1,2) average + 2 Newton & 2.92e-4 & 6.378 & 0.951 & 1.59e-5 & 6.301 & 0.952 & 9.30e-7 & 6.342 & 0.953 \\
(2,0) half-block jackknife & 6.674 & 9.231 & 0.824 & 1.888 & 6.568 & 0.943 & 0.958 & 6.404 & 0.950 \\
(2,1) half-block + 1 Newton & 0.019 & 6.378 & 0.951 & 0.001 & 6.301 & 0.951 & 3.22e-4 & 6.342 & 0.953 \\
(2,2) half-block + 2 Newton & 1.67e-7 & 6.378 & -- & 1.40e-9 & 6.301 & -- & 7.37e-11 & 6.342 & -- \\
(3,0) quarter-block & 13.791 & 15.259 & -- & 1.925 & 6.571 & -- & 1.281 & 6.458 & -- \\
(3,1) quarter-block + 1 Newton & 0.984 & 6.440 & 0.949 & 0.001 & 6.301 & 0.952 & 5.11e-4 & 6.342 & 0.953 \\
analytic Cox--Snell + 1 Newton & 5.91e-4 & 6.378 & 0.951 & 2.62e-4 & 6.301 & 0.951 & 1.30e-4 & 6.342 & 0.953 \\
CSL, 1 round & 6.304 & 8.966 & -- & 2.097 & 6.631 & -- & 0.718 & 6.378 & -- \\
CSL, 2 rounds & 2.003 & 6.709 & -- & 0.431 & 6.314 & -- & 0.101 & 6.344 & -- \\
AEE (Lin--Xi) & 41.193 & 41.683 & -- & 20.707 & 21.655 & -- & 10.393 & 12.159 & -- \\
(2,1) stale Hessian & 0.209 & 6.382 & -- & 0.031 & 6.301 & -- & 0.008 & 6.342 & -- \\
centralized MLE & 0 & 6.378 & 0.951 & 0 & 6.301 & 0.952 & 0 & 6.342 & 0.953 \\
\bottomrule
\end{tabular}
\begin{tablenotes}\footnotesize
\item 1000 paired replications per column. gap $=$ root mean square of $\sqrt N\,\lVert\tilde\theta - \widehat\theta_N\rVert$; RMSE $=$ root mean square of $\sqrt N\,\lVert\tilde\theta - \theta_0\rVert$; 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 $\approx 0.007$.
\item Fraction of workers with all local fits valid (orders 2 / 3): $m=256$: 1.000 / 1.000; $m=512$: 1.000 / 1.000; $m=1024$: 1.000 / 1.000.
\end{tablenotes}
\end{threeparttable}
\end{table}