This package provides functions to find solutions to penalized quantile regression problems. Throughout this package, the estimated coefficients are the minimizers of the penalized quantile regression objective function: $$\beta = \frac{1}{n}\sum_{i=1}^{n} \rho_{\tau}(y_i - x_i^T \beta) + \sum_{j=1}^{p} p_{\lambda}(|\beta_{j}|)$$, where $$\rho_{\tau}(u) = u(\tau - I(u < 0))$$. This package can handle three different penalty functions with \(\lambda > 0\):
LASSO: $$p_{\lambda}(|\beta_j|) = \lambda|\beta_j|$$
SCAD: $$p_{\lambda}(|\beta_j|) = \lambda|\beta_j|I(0\leq |\beta_j| < \lambda) + \frac{a\lambda|\beta_j|-(\beta_j^2+\lambda^2)/2}{a-1}I(\lambda \leq |\beta_j| \leq a\lambda) + \frac{(a+1)\lambda^2}{2}I(|\beta_j| > a\lambda),$$ for \(a > 2\)
MCP: $$p_{\lambda}(|\beta_j|) = \lambda(|\beta_j|-\frac{\beta_{j}^2}{2a\lambda})I(0 \leq |\beta_j| \leq a\lambda) + \frac{a\lambda^2}{2}I(|\beta_j| > a\lambda),$$ for \(a > 1\).