## 4 Simulation Studies and Numerical Justifications
### Performance of RASMR, Algorithm 1
Nevertheless, based on our experience, most of the RASMR split requirements are not fulfilled, thus the resulting final number of data blocks of RASMR is only slightly larger than that of SMR. Similar to [43], we calculate the corresponding RMSE from the full data estimate \([\sum_{i=1}^{7}(\hat{\beta}_{i}-\hat{\beta}_{i})^{2}/7]^{1/2}\) and the RMSE from the true parameter values \([\sum_{i=1}^{7}(\hat{\beta}_{i}-\hat{\beta}_{i})^{2}/7]^{1/2}\), respectively. The simulation follows the same settings as in Example 1, and the comparisons are among MR, SMR with \(T=3\), RASMR with \(T=3\), and DC with \(K=1000\). The results are listed in Table 6. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Simulation** & \multicolumn{5}{c}{**Poisson Regression, \(K\)-Means (\(K\) = 1000)**} \\ \hline
**Representatives** & **MR** & **Percentage of NAs** & **Original SMR** & **Percentage of NAs** & **RASMR** & **Percentage of NAs** \\ \hline
**pre-norm** & 37.5 (12.9) & 0\% & 13.2 (10.2) & 0\% & 2.0 (0.5) & 0\% \\
**nzNormal** & 37.5 (12.9) & 0\% & 23.6 (15.7) & 3\% & 0.2 (\(3.6\times 10^{-2}\)) & 0\% \\
**ueNormal** & 82.5 (25.9) & 0\% & 9.5 (7.2) & 69\% & \(8.5\times 10^{-2}\) (\(2.3\times 10^{-2}\)) & 0\% \\
**nzNormal** & 49.9 (20.7) & 0\% & 29.5 (17.7) & 1\% & 0.07 (0.1) & 0\% \\
**\(T_{3}\)** & 298 (458) & 0\% & 97.2 (144) & 10\% & 31.3 (79.1) & 0\% \\ EXP & 31.2 (0.7) & 0\% & 6.2 (1.0) & 0\% & \(1.9\times 10^{-4}\) (\(5.8\times 10^{-5}\)) & 0\% \\ BETA & 0.5 (0.2) & 0\% & 2.3 (0.3) & 0\% & \(1.2\times 10^{-7}\) (\(1.8\times 10^{-9}\)) & 0\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Average (std) of RMSEs (\(10^{-3}\)) over 100 simulations for the Poisson regression. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Simulation** & \multicolumn{5}{c}{**Gamma Regression, \(K\)-Means (\(K\) = 1000)**} \\ \hline
**Representatives** & **MR** & **Percentage of NAs** & **Original SMR** & **Percentage of NAs** & **RASMR** & **Percentage of NAs** \\ \hline
**Beta** & 7.4 (0.7) & 0\% & 5.8 (1.2) & 11\% &
\begin{tabular}{c} 2.0 \(\times 10^{-2}\) \\ (0.2) \\ \end{tabular} & 0\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Average (std) of RMSEs (\(10^{-3}\)) over 100 Simulations for Gamma Regression. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Simulation** & \multicolumn{5}{c}{**Binary Classification, \(K\)-Means (\(K\) = 1000)**} & **True Link Function = Logit** \\ \hline
**Representatives** & \multicolumn{5}{c}{**SMR (\(T\) = 3)**} & \multicolumn{5}{c}{**RASMR (\(T\) = 3)**} \\ \hline Setup & Logit & Cloglog & Probit & Cauchit & Logit & Cloglog & Probit & Cauchit \\ \hline mzNormal & 6.04 & 8.95 & 11.09 & 6.51 & 7.32 & 9.75 & 12.95 & 7.63 \\ nzNormal & 5.99 & 8.90 & 12.82 & 6.22 & 6.60 & 9.82 & 13.65 & 6.70 \\
**ueNormal** & 6.75 & 10.33 & 13.55 & 7.14 & 7.68 & 10.38 & 15.82 & 7.79 \\ mixNormal & 5.90 & 8.52 & 11.26 & 6.22 & 6.63 & 9.52 & 12.64 & 7.02 \\ \(T_{3}\) & 6.32 & 8.43 & 8.10 & 6.36 & 6.77 & 9.06 & 8.50 & 7.00 \\ EXP & 5.66 & 7.98 & 10.51 & 6.24 & 6.55 & 9.20 & 12.11 & 6.90 \\ BETA & 5.66 & 7.96 & 10.73 & 6.36 & 6.51 & 9.10 & 12.57 & 7.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Average CPU time (seconds) over 100 simulations for binary classification. According to Table 6, with the same number of iterations (\(T=3\)), RASMR still outperforms MR, SMR, and DC. When the comparison is made with respect to the true parameter values, RASMR performs almost as well as the full data estimate with negligible differences. Recall that the data contain the entire information for estimating the parameters and the RASMR approach achieves its theoretical extreme in terms of estimation accuracy in this case.