# Error-Function-Based Penalized Quantile Regression in the Linear Mixed Model
## 3 Monte Carlo Simulation
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\boldsymbol{\tau}\) & **Sample Size** & **Correlation Coefficient** & \(\boldsymbol{\sigma}\) & **PE (SD)** & **TP** & **TN** \\ \hline \multirow{8}{*}{0.25} & \multirow{8}{*}{n = 30} & \(\rho=0.5\) & 0.2 & 0.110 (0.101) & 3 & 4.36 \\  & & & 2 & 0.290 (0.201) & 3 & 4.36 \\  & & \(\rho=0.8\) & 0.2 & 0.123 (0.101) & 3 & 3.98 \\  & & & 2 & 0.223 (0.138) & 3 & 3.72 \\ \cline{2-6}  & & \(\rho=0.5\) & 0.2 & 0.067 (0.071) & 3 & 3.89 \\  & & & 2 & 0.120 (0.092) & 3 & 3.68 \\  & & \(\rho=0.8\) & 0.2 & 0.073 (0.066) & 3 & 3.55 \\  & & & 2 & 0.103 (0.085) & 3 & 3.33 \\ \hline \multirow{8}{*}{0.5} & \multirow{8}{*}{n = 30} & \(\rho=0.5\) & 0.2 & 0.097 (0.088) & 3 & 3.97 \\  & & & 2 & 0.261 (0.185) & 3 & 4.48 \\  & & \(\rho=0.8\) & 0.2 & 0.088 (0.081) & 3 & 3.87 \\  & & & 2 & 0.191 (0.123) & 3 & 3.81 \\ \cline{2-6}  & & \(\rho=0.5\) & 0.2 & 0.059 (0.046) & 3 & 4.01 \\  & & & 2 & 0.119 (0.080) & 3 & 4.03 \\  & & \(\rho=0.8\) & 0.2 & 0.074 (0.064) & 3 & 3.27 \\  & & & 2 & 0.096 (0.079) & 3 & 3.33 \\ \hline \multirow{8}{*}{0.75} & \multirow{8}{*}{n = 30} & \(\rho=0.5\) & 0.2 & 0.130 (0.105) & 3 & 4.01 \\  & & & 2 & 0.321 (0.229) & 3 & 4.11 \\  & & \(\rho=0.8\) & 0.2 & 0.110 (0.094) & 3 & 4.10 \\  & & & 2 & 0.204 (0.129) & 3 & 3.81 \\ \cline{1-1} \cline{2-6}  & & \(\rho=0.5\) & 0.2 & 0.064 (0.055) & 3 & 3.85 \\  & & & 2 & 0.122 (0.091) & 3 & 3.70 \\  & & \(\rho=0.8\) & 0.2 & 0.080 (0.068) & 3 & 3.47 \\  & & & 2 & 0.117 (0.091) & 3 & 3.53 \\ \hline \end{tabular}
\end{table} 
Table 1: PE, TP, and TN for different correlation coefficients and sample sizes. Figure 2: PE for different correlation coefficients and sample size. Consider different SNR \(v=1\) and \(v=2\), and define three different random-effect matrices, \(D_{1}=diag(1,1,0,0)\), \(D_{2}=diag(2,2,0,0)\) and \(D_{3}=diag(3,3,0,0)\), to investigate the effects of different SNRs and random effects on the model. Take \(\tau=0.5\), \(n=30\) and \(\rho=0.5\). Table 2 indicates the average values of PE, TP, and TN in 100 simulations. In Table 2, as the SNR becomes greater and the influence of random effects increases, PE and SD increase gradually, while TN decreases slightly. If the SNR becomes greater, the model is less accurate and loses stability. In this case, it is more likely to select incorrect variables. However, we observe that even in the case \(v=2\) and \(D=D_{3}\), there are still relatively low errors for the model, and most incorrect variables are excluded. In general, an increase in SNR and random effects significantly impacts the coefficient estimation and variable selection. Meanwhile, the model is robust and effective in the process. To investigate the performance of DERF-QR and other methods, we discuss the following four scenarios with different error distributions. \(\varepsilon_{ij}\sim N(0,1)\), \(\varepsilon_{ij}\sim t(3)\), \(\varepsilon_{ij}\sim Cauchy(1,0)\) and \(\varepsilon_{ij}\sim 0.8N(0,1)+0.2N\big{(}0,5^{2}\big{)}\), which denotes \(\varepsilon_{ij}\sim\) mixnorm. We would make comparisons among the following four methods: Quantile Regression without penalization (QR), Doubly penalized Lasso regularized Quantile Regression (DLQR), Doubly penalized SCAD regularized Quantile Regression (DSQR), and the DERF-QR model with the parameter \(\sigma=0.2\) proposed in this paper. Consider two distinct scenarios as follows:  
\[ \begin{split}\text{\emph{sparse model}}&\beta=(3,1,6,0, 0,0,0,0)^{\text{T}},\\ \text{\emph{dense model}}&\beta=(0.75,0.75,0.75,0.75,0.7 5,0.75,0.75,0.75)^{\text{T}}.\end{split} \]  
Repeat 100 times, and take \(\tau=0.5\), \(n=30\) and \(\rho=0.5\). Tables 3 and 4 depict the results, and Figures 3 and 4 show the boxplots of PE for these two models. In the dense model, DERF-QR still performs the best among the four methods. We could select the correct variables from all by these four methods. In most cases, the Cauchy distribution shows smaller errors, while the mixnorm distribution results in relatively greater errors.