# Error-Function-Based Penalized Quantile Regression in the Linear Mixed Model
## 3 Monte Carlo Simulation  
The simulated data are generated from the following model:  
\[ \begin{split}\gamma_{ij}=\beta_{0}+x_{ij}^{\mathrm{T}}\beta+z_{ij}^{ \mathrm{T}}a_{i}+v\epsilon_{ij},\quad i=1,2,\ldots,n,\quad j=1,2,\ldots,m\end{split}  \tag{12}\]  
Here, let \(x_{ij}=\big{(}x_{ij1},\,\ldots,x_{ij8}\big{)}^{\mathrm{T}}\), \(\beta_{0}=0\), \(\beta=(3,1,6,0,0,0,0,0)^{\mathrm{T}}\), \(n=30\), and \(m=10\). \(X_{k}=(x_{11k},\ldots,x_{mmk})^{\mathrm{T}}\sim N_{nm}(0,I)\) is the covariate vector, and \(\rho^{|l-k|}\) is the correlation between \(X_{l}\) and \(X_{k}\). For random effects, define \(z_{ij}=\big{(}1,x_{ij1},x_{ij2},x_{ij3}\big{)}^{\mathrm{T}}\) for the first three variables, and let \(\alpha_{i}=(\alpha_{i1},\alpha_{i2},\alpha_{i3},\alpha_{i4})^{\mathrm{T}}\sim N (0,D)\), where \(D=diag(1,1,0,0)\). \(v\) denotes the signal-to-noise ratio (SNR). In the following, we introduce \(\mathrm{PE}\), \(\mathrm{TP}\), and \(\mathrm{TN}\) to evaluate the variable selection and parameter estimation in the model. (1) Prediction Error (PE):  
\[\mathrm{PE}=\big{(}\hat{\beta}-\beta\big{)}^{\mathrm{T}}\Sigma\big{(}\hat{ \beta}-\beta\big{)}, \tag{13}\]  
where \(\hat{\beta}\) is the estimate of \(\beta\), \(\sum=\left(\rho^{|l-k|}\right)_{8\times 8}\). (2) True positives (TP): The number of variables selected correctly. (3) True negatives (TN): The number of variables excluded correctly. We show the performance of the DERF-QR method for different correlation coefficients and different sample sizes, that is, there are four scenarios, as follows: (1) \(n=30\), \(\rho=0.5\), (2) \(n=30\), \(\rho=0.8\), (3) \(n=50\), \(\rho=0.5\), (4) \(n=50\), \(\rho=0.8\) at quantiles \(\tau=0.25\), \(0.5\), and \(0.75\), respectively. In addition, we discuss the cases \(\sigma=0.2\) and \(2\), respectively. Here, \(\epsilon_{ij}\sim N(0,1)\) and \(v=1\). We select the penalty parameters by EBIC and take \(\gamma_{p}=1\). We take the coefficient \(\beta_{l}=0\) if \(|\beta_{l}|<10^{-4}\), and the same for \(\alpha_{i}\). Table 1 reports the mean values of \(\mathrm{PE}\), standard deviation (SD), and the mean values of \(\mathrm{TP}\) and \(\mathrm{TN}\) based on \(100\) replications. The simulation shows that we could select the right variable by DERF-QR in all experimental conditions in Table 1. Also, \(\mathrm{PE}\) decreases and \(\mathrm{TN}\) decreases slightly when sample size \(n\) increases. It is observed that stronger correlation \(\rho=0.8\) yields smaller \(\mathrm{PE}\) values, compared with \(\rho=0.5\). The DERF-QR method in strong-correlation scenarios shows robust performance due to its adaptive penalty weights. This improves coefficient estimation accuracy under strong variable correlations. Parameter \(\sigma\) significantly influences the result. The \(\mathrm{PE}\) value of \(\sigma=2\) is greater than that of \(\sigma=0.2\). Additionally, we observe that the model is less sensitive to variations of the parameter \(\sigma\) at the median quantile. The median quantile regression also yields a smaller \(\mathrm{PE}\) value, compared with other quantiles. The simulation results are presented in Figure 2.  
Figure 1: Common penalized proximal operators.