## 4 Simulation Studies and Numerical Justifications
### Performance of RASMR with the Delta Ratio Split, Algorithm 2  
In Section 2.4, we develop Algorithm 2 for estimating the maximum likelihood better, especially for non-Gaussian covariates or predictors. To justify the improvement of Algorithm 2 against RASMR, in this section, we employ the same simulation setting as described in Section 4.1 (see Example 1). For illustration purposes, the threshold \(\delta_{0}\) for delta ratio split is chosen to be 0.05, 0.1, 0.5, and 1, respectively. We fit the logistic regression model using SMR, RASMR, and RASMR with the delta ratio split and various thresholds. The estimated maximum log-likelihood approaches the full data value as the threshold of the delta ratio decreases, except in the \(T_{3}\) case, where the improvement is negligible (see Figure S11 in the Supplementary Materials). One reason is that in the case of \(T_{3}\), the delta ratios are much smaller than those in other cases (see Figure S1 in the Supplementary Materials), and even 0.05 does not capture a good amount of blocks. To evaluate how the delta ratio split impacts the accuracy of parameter estimates, we choose \(K=100,500\), and 1000 for \(K\)-means clustering, respectively. With moderately large \(K\), such as \(K=500\) or 1000, RASMR with a delta ratio split performs roughly the same as the original RASMR (see Figure S12 in the Supplementary Materials), and both RASMR and RASMR with a delta ratio split outperform SMR. Nevertheless, when \(K\) is as small as 100, the RASMR algorithms may fail for some data structures, such as nzNormal (where the data have imbalanced responses), \(T_{3}\) (where the data distribute with heavy tails), or mixNormal (where the data come from a mixture of two distributions). In Table 7, we list the performance of RASMR with the delta ratio split and \(\delta_{0}=0.1\) when the link function is misspecified. Comparing Table 7 with Table 2, we can see that RASMR with the delta ratio split (Algorithm 2) is, in general, better than or comparable with Algorithm 1 in terms of estimation accuracy. The improvements are significant in the unbalanced or heavy-tailed cases, such as nzNormal with cauchit link, and ueNormal with cloglog or cauchit links (see Section S5 in the Supplementary Materials for more details). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Simulation**} & \multicolumn{6}{c}{**Binary Classification, \(K\)-Means (\(K=1000\)), True Link Function = Logit**} \\ \hline \multicolumn{1}{c}{**Benchmark**} & \multicolumn{3}{c}{**RMSE from Full Data Estimate**} & \multicolumn{6}{c}{**RMSE from True Parameter Value**} \\ \cline{2-9} \multicolumn{1}{c}{**Setup–Methods**} & \multicolumn{3}{c}{**Representative Approaches**} & \multicolumn{1}{c}{**DC**} & \multicolumn{1}{c}{**Full Data**} & \multicolumn{3}{c}{**Representative Approaches**} & **DC** \\ \cline{2-9} \multicolumn{1}{c}{} & **MR** & **SMR (\(T=3\))** & **RASMR (\(T=3\))** & **MR** & **SMR (\(T=3\))** & **RASMR (\(T=3\))** & **RSMR (\(T=3\))** & **DS** \\ \hline mzNormal & 17.98(0.31) & 1.92(0.62) & 0.054(0.018) & 6.93(0.12) & 3.72(1.08) & 18.04(0.12) & 4.17(1.16) & 3.272(1.08) & 7.90(1.04) \\ nzNormal & 14.30(0.70) & 4.53(1.8) & 0.29(0.085) & 20.20(0.35) & 7.232(0.05) & 16.04(1.74) & 8.58(2.02) & 7.242(0.7) & 21.49(1.64) \\ neNormal & 211.17(1.44) & 3.83(1.57) & 0.72(0.018) & 13.12(0.26) & 2.11(0.82) & 211.17(1.35) & 4.41(1.77) & 2.22(0.84) & 13.24(1.24) \\ mixNormal & 17.37(0.33) & 2.79(0.83) & 0.14(0.043) & 11.20(0.20) & 4.96(1.33) & 17.94(1.05) & 5.91(1.48) & 4.96(1.33) & 12.09(1.15) \\ \(T_{3}\) & 12.23(3.13) & 1.13(0.32) & 1.89(0.74) & 12.06(0.34) & 16.00(0.443) & 20.51(5.44) & 20.03(5.54) & 16.03(5.41) & 19.63(3.76) \\ EXP & 12.49(1.5) & 6.58(0.82) & 0.014(0.0013) & 16.88(0.31) & 6.18(1.67) & 14.52(1.8) & 9.25(2.02) & 6.18(1.67) & 18.25(2.30) \\ BETA & 3.03(0.80) & 2.34(0.68) & 0.00031(0.00090) & 5.92(0.20) & 7.49(2.38) & 7.89(2.30) & 7.73(2.34) & 7.49(2.38) & 9.31(2.54) \\ \hline \hline \end{tabular}
\end{table}