## 3 Model Selection and Variable Selection Using RASMR  
### Information-Based Criteria and Model Selection  
During the process of RASMR parameter estimation by Algorithms 1 and 2, two very useful side products are also generated, namely the response representative \(\tilde{y}_{k}\) and the predictor representative \(\tilde{\mathbf{X}}_{k}\). The quadruple \((n_{k},\tilde{y}_{k},\tilde{\mathbf{X}}_{k},\tilde{\beta})\) provides information for approximating the maximum likelihood, which is a key component for information-based criteria, such as the Akaike information criterion (AIC, [54; 55]) and Bayesian information criterion (BIC, [55]). Recall that the log-likelihood function \(l(\mathbf{\beta};\mathbf{y},\mathbf{X})\) through the full dataset \(\mathbf{y}=(y_{1},\ldots,\allowbreak y_{N})^{T}\) and \(\mathbf{X}=(\mathbf{X}_{1},\ldots,\mathbf{X}_{N})^{T}\) attains its maximum at the MLE \(\tilde{\mathbf{\beta}}\). We denote the log-likelihood approximated through the RASMR representative quadruple set \(\{(n_{k},\tilde{y}_{k},\tilde{\mathbf{X}}_{k},\tilde{\beta}),\)\(k=1,\ldots,K\}\) by \(l(\tilde{\mathbf{\beta}};\tilde{\mathbf{y}},\tilde{\mathbf{X}})\), where \(\tilde{\mathbf{y}}=(\tilde{y}_{1},\ldots,\tilde{y}_{K})^{T}\), and \(\tilde{\mathbf{X}}=(\tilde{\mathbf{X}}_{1},\ldots,\tilde{\mathbf{X}}_{K})^{T}\). Denoting \(\tilde{\Delta}=\max_{k}\max_{i\in I_{k}}\|\mathbf{X}_{i}-\tilde{\mathbf{X}}_ {k}\|\), the maximum Euclidean distance of predictor vectors away from their corresponding predictor representatives across all data blocks, the following theorem guarantees the consistency of the estimated log-likelihood \(l(\tilde{\mathbf{\beta}};\mathbf{y},\mathbf{X})\), as \(\tilde{\Delta}\) goes to zero. **Theorem 1**. : _Suppose the log-likelihood function \(l(\mathbf{\beta};\mathbf{y},\mathbf{X})\) is twice differentiable and strictly concave on a compact set \(C\subset\mathcal{R}^{P}\), with its maximum not located on the boundary of \(C\). Suppose \(\tilde{\mathbf{\beta}}\) is the estimate obtained by Algorithms 1 or 2, and \(\tilde{y}_{k}=n_{k}^{-1}\sum_{i\in I_{k}}y_{i}+O(\tilde{\Delta})\) for each \(k\), then \(l(\tilde{\mathbf{\beta}};\mathbf{y},\mathbf{X})-l(\tilde{\mathbf{\beta}};\mathbf{y}, \mathbf{X})=O(\tilde{\Delta}^{1/2})\) as \(\tilde{\Delta}\to 0\)._  
The proof of Theorem 1, as well as all other proofs in this paper, has been relegated to Section S6 of the Supplementary Materials. We denote the AIC and BIC values calculated based on \(l(\tilde{\mathbf{\beta}};\mathbf{y},\mathbf{X})\) as \(\widetilde{AIC}\) and \(\widetilde{BIC}\), respectively. As a direct conclusion of Theorem 1, we have the following corollary:  
**Corollary 1**. : _For a generalized linear model under the same technical conditions of Theorem 1, the values of \(\widetilde{AIC}\) and \(\widetilde{BIC}\) converge to the original values of AIC and BIC, respectively, as \(\tilde{\Delta}\to 0\). Moreover, \(\widetilde{AIC}=AIC+O(\tilde{\Delta}^{1/2})\), and \(\widetilde{BIC}=BIC+O(\tilde{\Delta}^{1/2})\)._The computation of \(l(\hat{\mathbf{\beta}};\hat{\mathbf{y}},\hat{\mathbf{X}})\) only requires the \(K\) representatives rather than the whole dataset. Its accuracy not only depends on the estimate \(\hat{\mathbf{\beta}}\) of the parameters but also on the representatives \(\hat{\mathbf{y}}\) and \(\hat{\mathbf{X}}\). The next theorem confirms that the estimated log-likelihood \(l(\hat{\mathbf{\beta}};\hat{\mathbf{y}},\hat{\mathbf{X}})\) based on RASMR algorithms is consistent as \(\tilde{\Delta}\) goes to zero. Under the same technical conditions of Theorem 1, \(l(\hat{\mathbf{\beta}};\hat{\mathbf{y}},\hat{\mathbf{X}})-l(\hat{\mathbf{\beta}};\mathbf{y},\mathbf{X})= O(\tilde{\Delta}^{1/2})\), as \(\tilde{\Delta}\to 0\). We denote the AIC and BIC calculated with \(l(\hat{\mathbf{\beta}};\hat{\mathbf{y}},\hat{\mathbf{X}})\) as \(\widehat{AIC}\) and \(\widehat{BIC}\), respectively.