\begin{table}[p]
\centering\scriptsize
\begin{threeparttable}
\caption{Monte Carlo, logit design, $p = 3$, $N = 1,048,576$: scaled gap to $\widehat\theta_N$, scaled RMSE and 95\% coverage by local sample size $m$.}\label{tab:mc_logit_p3}
\begin{tabular}{@{}lrrrrrrrrrrrr@{}}
\toprule
Estimator & \multicolumn{3}{c}{$m = 32$, $K = 32,768$} & \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 & RMSE & cov & gap & RMSE & cov & gap & RMSE & cov & gap & RMSE & cov \\
\midrule
(1,0) plain average & 286.566 & 286.648 & 0.000 & 29.019 & 29.330 & 0.008 & 6.969 & 7.892 & 0.572 & 3.463 & 5.064 & 0.860 \\
(1,1) average + 1 Newton & 2689.961 & 2689.977 & 0.331 & 0.318 & 3.757 & 0.951 & 0.018 & 3.744 & 0.951 & 0.004 & 3.751 & 0.957 \\
(1,2) average + 2 Newton & 8.41e+6 & 8.41e+6 & 0.945 & 3.76e-5 & 3.751 & 0.952 & 1.21e-7 & 3.744 & 0.951 & 6.47e-9 & 3.751 & 0.957 \\
(2,0) half-block jackknife & 1.45e+4 & 1.45e+4 & 0.000 & 3.782 & 5.308 & 0.829 & 0.514 & 3.775 & 0.949 & 0.338 & 3.769 & 0.958 \\
(2,1) half-block + 1 Newton & 2.04e+8 & 2.04e+8 & 0.004 & 0.006 & 3.751 & 0.952 & 1.09e-4 & 3.744 & 0.951 & 4.55e-5 & 3.751 & 0.957 \\
(2,2) half-block + 2 Newton & 1.18e+24 & 1.18e+24 & -- & 1.39e-8 & 3.751 & -- & 1.08e-9 & 3.744 & -- & 1.08e-9 & 3.751 & -- \\
(3,0) quarter-block & 2.50e+4 & 2.50e+4 & -- & 79.768 & 79.856 & -- & 0.678 & 3.792 & -- & 0.473 & 3.781 & -- \\
(3,1) quarter-block + 1 Newton & 1.52e+9 & 1.52e+9 & 0.008 & 144.811 & 144.911 & 0.947 & 1.73e-4 & 3.744 & 0.951 & 8.65e-5 & 3.751 & 0.957 \\
analytic Cox--Snell + 1 Newton & 4.62e+117 & 4.62e+117 & 0.457 & 1.76e-4 & 3.751 & 0.952 & 4.49e-5 & 3.744 & 0.951 & 2.13e-5 & 3.751 & 0.957 \\
CSL, 1 round & 57.743 & 57.803 & -- & 4.334 & 5.683 & -- & 0.510 & 3.790 & -- & 0.170 & 3.754 & -- \\
CSL, 2 rounds & 50.483 & 50.642 & -- & 1.233 & 3.960 & -- & 0.064 & 3.744 & -- & 0.014 & 3.751 & -- \\
AEE (Lin--Xi) & 102.585 & 102.557 & -- & 26.847 & 27.041 & -- & 6.829 & 7.809 & -- & 3.423 & 5.119 & -- \\
(2,1) stale Hessian & 1.24e+4 & 1.24e+4 & -- & 0.102 & 3.753 & -- & 0.004 & 3.744 & -- & 0.001 & 3.751 & -- \\
centralized MLE & 0 & 3.806 & 0.950 & 0 & 3.751 & 0.952 & 0 & 3.744 & 0.951 & 0 & 3.751 & 0.957 \\
\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): $m=32$: 0.961 / 0.234; $m=64$: 1.000 / 0.924; $m=128$: 1.000 / 1.000; $m=256$: 1.000 / 1.000; $m=512$: 1.000 / 1.000; $m=1024$: 1.000 / 1.000.
\end{tablenotes}
\end{threeparttable}
\end{table}