Journal of Statistical Planning and Inference
Given any quantile level \(\tau\) \((0 < \tau < 1)\), consider the \(\tau\)-th varying coefficient quantile regression
\[Y = \sum^p_{j=1}g_{j, \tau}(U)X^{(j)} + \varepsilon,\]where \(U \in [0,1]\) is an index variable, \(g_\tau = (g_{1,\tau(u)}, \ldots, g_{p,\tau(u)})^\top\) is a vector of coefficient functions, and the conditional \(\tau\)-th quantile of a random error \(\varepsilon\) given \((U, \{X^{(j)}\}_{1\le j \le p})\) is zero.
Suppose that the underlying coefficient functions \(g_k(u)\), \(k=1,\ldots, p\), are smooth. Then, the smooth function \(g_k(\cdot)\) can be approximated in a locally linear manner, i.e., \(g_k(U) \approx g_k(u) + g'_k(u)(U-u)\) for any \(U\) close to \(u\). That is, given \(0 \le u \le 1,\) estimate \(g_k(u)\) and \(g_k'(u)\); by minimizing
\[\sum^n_{i=1}\rho_\tau\left(Y_i - \sum^p_{k=0}X_i^{(k)}(a_k + b_k(U_i - u))\right)K_h(u - U_i),\]with respect to \(a_k, b_k \in \mathbb{R}\), where \(\rho_\tau(t) = t(\tau - I(t \le 0))\) is the check loss function. As a result, we can consider the following loss function for all \(u\) by integrating such that
\[Q({\bf a}, {\bf b}) = \frac{1}{n}\int^1_0 \sum^n_{i=1}\rho_\tau\left(Y_i - \sum^p_{k=0}X_i^{(k)}(a_k + b_k(U_i - u))\right)\times K_h(u - U_i)du.\]Furthermore, using the following penalized criterion
\[Q({\bf a}, {\bf b}) + \sum^p_{j=1}w_j\sqrt{\|a_j\|^2 + h^2\|b_j\|^2} + \sum^p_{j=1}v_j\sqrt{\|a_j\|^2_c + h^2\|b_j\|^2}\] \[\text{subject to } \|a_j\|_\infty \vee \|hb_j\|_\infty \le M_1, \sqrt{\|a_j^{(2)}\|^2 + \|hb_j^{(2)}\|^2} \le M_2\]for all \(j=1,\ldots,p\) for some positive constant \(M_1\) and \(M_2\).