\begin{table}[p]
\centering\scriptsize\setlength{\tabcolsep}{2.6pt}
\begin{threeparttable}
\caption{Monte Carlo, logit illcond 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_illcond_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}{$m = 256$, $K = 4,096$} & \multicolumn{3}{c}{$m = 512$, $K = 2,048$} \\
 & gap & median & cov & gap & median & cov & gap & median & cov & gap & median & cov \\
\midrule
(1,0) plain average & 210.693 & 210.817 & 0.043 & 91.092 & 90.942 & 0.196 & 43.123 & 42.945 & 0.361 & 21.195 & 21.063 & 0.580 \\
(1,1) average + 1 Newton & 11.713 & 11.700 & 0.784 & 2.072 & 2.069 & 0.949 & 0.452 & 0.451 & 0.952 & 0.107 & 0.106 & 0.959 \\
(1,2) average + 2 Newton & 0.033 & 0.033 & 0.947 & 0.001 & 0.001 & 0.948 & 4.90e-5 & 4.88e-5 & 0.951 & 2.71e-6 & 2.67e-6 & 0.959 \\
(2,0) half-block jackknife & 2.71e+4 & 7151.921 & 0.005 & 47.520 & 31.096 & 0.466 & 10.088 & 9.224 & 0.900 & 5.965 & 5.276 & 0.953 \\
(2,1) half-block + 1 Newton & 6.94e+7 & 4.21e+4 & 0.004 & 1.181 & 0.210 & 0.949 & 0.011 & 0.010 & 0.951 & 0.002 & 0.001 & 0.959 \\
(2,2) half-block + 2 Newton & 3.79e+18 & 5.48e+6 & -- & 0.003 & 1.00e-5 & -- & 2.34e-8 & 1.47e-8 & -- & 4.98e-10 & 4.14e-10 & -- \\
(3,0) quarter-block & 2.34e+5 & 1.94e+4 & -- & 5229.071 & 2317.092 & -- & 13.224 & 11.649 & -- & 8.334 & 7.065 & -- \\
(3,1) quarter-block + 1 Newton & 3.59e+10 & 4.72e+5 & 0.015 & 8.31e+4 & 2202.654 & 0.008 & 0.010 & 0.006 & 0.951 & 0.002 & 0.001 & 0.959 \\
(2,1) valid workers only & 15.329 & 15.326 & -- & 0.214 & 0.209 & -- & -- & -- & -- & -- & -- & -- \\
(3,1) valid workers only & 10.267 & 9.317 & -- & 0.450 & 0.433 & -- & -- & -- & -- & -- & -- & -- \\
analytic Cox--Snell + 1 Newton & 0.025 & 0.021 & 0.947 & 0.002 & 0.002 & 0.948 & 8.32e-4 & 5.45e-4 & 0.951 & 3.88e-4 & 2.64e-4 & 0.959 \\
CSL, 1 round & 245.222 & 195.955 & -- & 67.596 & 59.366 & -- & 21.791 & 18.980 & -- & 7.123 & 6.205 & -- \\
CSL, 2 rounds & 234.725 & 124.589 & -- & 29.868 & 20.046 & -- & 5.641 & 3.868 & -- & 1.101 & 0.844 & -- \\
AEE (Lin--Xi) & 154.527 & 154.479 & -- & 78.136 & 78.122 & -- & 39.644 & 39.631 & -- & 20.046 & 20.045 & -- \\
(2,1) stale Hessian & 2.50e+4 & 2269.895 & -- & 4.833 & 2.341 & -- & 0.383 & 0.348 & -- & 0.114 & 0.101 & -- \\
centralized MLE & 0 & -- & 0.947 & 0 & -- & 0.948 & 0 & -- & 0.951 & 0 & -- & 0.959 \\
\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): $m=64$: 0.973 / 0.022; $m=128$: 1.000 / 0.947; $m=256$: 1.000 / 1.000; $m=512$: 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}