# Error-Function-Based Penalized Quantile Regression in the Linear Mixed Model
## 2 Model and Algorithm
Nevertheless, their non-convex and non-smooth nature poses significant challenges in solving the associated optimization problems. ERF regularization is a non-convex penalty function with smoothness, which increases flexibility in the optimization process and improves computational efficiency. Existing work has shown that this function reduces bias more effectively than most other non-convex penalties [16]. With appropriate parameters \(\sigma\), the ERF regularization term can closely approximate the \(L_{0}\) penalty, which directly controls the sparsity of the model and yields an optimal sparse solution. For any \(x\in\mathbb{R}^{n}\), we have  
\[J_{\sigma}(x) \rightarrow\|x\|_{1}, \text{if }\sigma\rightarrow+\infty,\] \[\frac{J_{\sigma}(x)}{\sigma} \rightarrow\frac{\sqrt{\pi}}{2}\|x\|_{0}, \text{if }\sigma\to 0^{+}.\]  
Moreover, [PERSON] provided the following four properties and their proofs [16]. (1) The error function \(\Phi(x)\) is smooth, and the following is its derivative:  
\[\frac{d}{dx}\Phi_{\sigma}(x)=\exp\biggl{(}-\Bigl{(}\frac{x}{\sigma}\Bigr{)}^{2} \biggr{)}. \tag{5}\]  
(2) For any \(x\in\mathbb{R}\), we have  
\[c\sqrt{1-e^{-ax^{2}}}\leq\Phi_{\sigma}(x)\leq c\sqrt{1-e^{-bx^{2}}}, \tag{6}\]  
where \(a=\frac{1}{\sigma^{2}}\), \(b=\frac{\pi}{(4\sigma)^{2}}\) and \(c=\frac{\sigma\sqrt{\pi}}{2}\). (3) \(J_{\sigma}\in\mathbb{R}_{+}^{n}\) is concave. For any \(t\in[0,1]\) and \(x,y\in\mathbb{R}_{+}^{n}\), we have  
\[Q\big{(}\tau\mid x_{ij},z_{ij},\alpha_{i}\big{)}=x_{ij}^{\mathrm{T}}\beta+z_{ ij}^{\mathrm{T}}\alpha_{i}, \tag{2}\]0  
(4) For any \(x,y\in\mathbb{R}^{n}\),  
\[Q\big{(}\tau\mid x_{ij},z_{ij},\alpha_{i}\big{)}=x_{ij}^{\mathrm{T}}\beta+z_{ ij}^{\mathrm{T}}\alpha_{i}, \tag{2}\]1  
These properties show that the error function is smooth and bounded. Equation (8) further confirms that the ERF regularization satisfies the triangle inequality. We employ the proximal operator to measure the bias of the penalty function [19]. For a function \(f\colon\mathbb{R}^{n}\to\mathbb{R}^{n}\cup\{+\infty\}\), the proximal operator \(\text{prox}(v)\colon\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}\) of the function \(f\) is defined as follows:  
\[Q\big{(}\tau\mid x_{ij},z_{ij},\alpha_{i}\big{)}=x_{ij}^{\mathrm{T}}\beta+z_{ ij}^{\mathrm{T}}\alpha_{i}, \tag{2}\]2  
In Figure 1, we plot the proximal operators for \(L_{1}\), \(L_{2}\), SCAD, and ERF with different \(\sigma\) at \(\mu=1\). The proximity of each proximal operator to the line \(y=x\) reflects the bias level, that is, a smaller deviation indicates lower bias. The bias of the ERF penalty increases as \(\sigma\) increases, while for all values \(\sigma\), it remains smaller than that of \(L_{1}\). Considering the non-smoothness of the quantile loss function and the non-convexity of the error function, we can transform Equation (4) into the following form by the iterative reweighted \(L_{1}\) algorithm [17]:  
\[Q\big{(}\tau\mid x_{ij},z_{ij},\alpha_{i}\big{)}=x_{ij}^{\mathrm{T}}\beta+z_{ ij}^{\mathrm{T}}\alpha_{i}, \tag{2}\]3 \[Q\big{(}\tau\mid x_{ij},z_{ij},\alpha_{i}\big{)}=x_{ij}^{\mathrm{T}}\beta+z_{ ij}^{\mathrm{T}}\alpha_{i}, \tag{2}\]4  
where the corresponding weights \(\omega_{\beta_{l}}\) are computed by differentiating \(\Phi_{\sigma}(|\beta_{l}|)\) for each \(\beta_{l}\). Specially, we have  
\[Q\big{(}\tau\mid x_{ij},z_{ij},\alpha_{i}\big{)}=x_{ij}^{\mathrm{T}}\beta+z_{ ij}^{\mathrm{T}}\alpha_{i}, \tag{2}\]5 \[Q\big{(}\tau\mid x_{ij},z_{ij},\alpha_{i}\big{)}=x_{ij}^{\mathrm{T}}\beta+z_{ ij}^{\mathrm{T}}\alpha_{i}, \tag{2}\]6  
We iteratively compute the weights and solve the optimization problem of Equation (9) until convergence is achieved. [PERSON] et al. proposed the proximal ADMM and sparse coordinate descent ADMM algorithms, and proved their global convergence [20]. These two methods are particularly effective for solving high-dimensional sparse quantile regression problems,and show high estimation accuracy and computational efficiency.