\begin{table}[p]
\centering\scriptsize
\begin{threeparttable}
\caption{Monte Carlo, logit design, $p = 8$, $N = 1,048,576$: scaled gap to $\widehat\theta_N$ (root mean square and median) and 95\% coverage by local sample size $m$.}\label{tab:mc_logit_p8}
\begin{tabular}{@{}lrrrrrrrrrrrr@{}}
\toprule
Estimator & \multicolumn{3}{c}{$m = 64$, $K = 16,384$} & \multicolumn{3}{c}{$m = 128$, $K = 8,192$} & \multicolumn{3}{c}{$m = 512$, $K = 2,048$} & \multicolumn{3}{c}{$m = 1024$, $K = 1,024$} \\
 & gap & median & cov & gap & median & cov & gap & median & cov & gap & median & cov \\
\midrule
(1,0) plain average & 233.195 & 226.356 & 0.000 & 94.927 & 94.925 & 0.000 & 21.491 & 21.476 & 0.148 & 10.589 & 10.579 & 0.633 \\
(1,1) average + 1 Newton & 31.483 & 23.333 & 0.099 & 3.909 & 3.908 & 0.916 & 0.195 & 0.194 & 0.952 & 0.047 & 0.047 & 0.953 \\
(1,2) average + 2 Newton & 2.275 & 0.225 & 0.942 & 0.006 & 0.006 & 0.956 & 1.59e-5 & 1.57e-5 & 0.952 & 9.30e-7 & 9.19e-7 & 0.953 \\
(2,0) half-block jackknife & 9.42e+4 & 1.81e+4 & 0.000 & 45.313 & 36.592 & 0.004 & 1.888 & 1.804 & 0.943 & 0.958 & 0.909 & 0.950 \\
(2,1) half-block + 1 Newton & 1.62e+11 & 2.23e+6 & 0.009 & 3.356 & 0.550 & 0.952 & 0.001 & 0.001 & 0.951 & 3.22e-4 & 2.26e-4 & 0.953 \\
(2,2) half-block + 2 Newton & 2.38e+24 & 1.52e+12 & -- & 0.125 & 1.26e-4 & -- & 1.40e-9 & 5.28e-10 & -- & 7.37e-11 & 2.05e-11 & -- \\
(3,0) quarter-block & 1.86e+5 & 8047.508 & -- & 2.85e+4 & 5947.678 & -- & 1.925 & 1.838 & -- & 1.281 & 1.218 & -- \\
(3,1) quarter-block + 1 Newton & 1.32e+12 & 1.56e+5 & 0.018 & 2.36e+9 & 4.44e+4 & 0.002 & 0.001 & 9.01e-4 & 0.952 & 5.11e-4 & 3.91e-4 & 0.953 \\
(2,1) valid workers only & -- & -- & -- & -- & -- & -- & -- & -- & -- & -- & -- & -- \\
(3,1) valid workers only & -- & -- & -- & -- & -- & -- & -- & -- & -- & -- & -- & -- \\
analytic Cox--Snell + 1 Newton & 9.07e+53 & 0.059 & 0.918 & 0.003 & 0.002 & 0.956 & 2.62e-4 & 1.90e-4 & 0.951 & 1.30e-4 & 9.20e-5 & 0.953 \\
CSL, 1 round & 76.119 & 66.001 & -- & 20.125 & 18.523 & -- & 2.097 & 1.973 & -- & 0.718 & 0.681 & -- \\
CSL, 2 rounds & 80.931 & 47.543 & -- & 10.636 & 7.749 & -- & 0.431 & 0.344 & -- & 0.101 & 0.083 & -- \\
AEE (Lin--Xi) & 163.596 & 163.559 & -- & 81.848 & 81.838 & -- & 20.707 & 20.711 & -- & 10.393 & 10.385 & -- \\
(2,1) stale Hessian & 9.31e+4 & 1.32e+4 & -- & 7.823 & 2.738 & -- & 0.031 & 0.030 & -- & 0.008 & 0.008 & -- \\
centralized MLE & 0 & -- & 0.945 & 0 & -- & 0.956 & 0 & -- & 0.952 & 0 & -- & 0.953 \\
\bottomrule
\end{tabular}
\begin{tablenotes}\footnotesize
\item 1000 paired replications per column. gap $=$ root mean square of $\sqrt N\,\lVert\tilde\theta - \widehat\theta_N\rVert$ 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 $\approx 0.007$ or less.
\item Fraction of workers with all local fits valid (orders 2 / 3): $m=64$: 0.930 / 0.009; $m=128$: 1.000 / 0.865; $m=256$: 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}