An algorithm for quantile smoothing splines

For p = 1 and infinity, Koenker, Ng and Portnoy (Statistical Data Analysis Based on the L(1) Norm and Related Methods (North-Holland, New York, 1992); Biometrika, 81 (1994)) proposed the tau th L(p) quantile smoothing spline, (g) over cap(tau,Lp), defined to solve (y is an element of Gp)min ''fidelity'' + lambda ''L(p) roughness'' as a simple, nonparametric approach to estimating the zth conditional quantile functions given 0 less than or equal to tau less than or equal to 1. They defined ''fidelity'' = (i=1)Sigma(n) rho(tau)(y(i)-g(x(i))) with rho(tau)(u)=(tau-I(u<0))u, ''L(1) roughness'' = (n-1)Sigma(i=1)\g'(x(i+1))-g'(x(i))\, ''L(infinity) roughness'' = max(x) g''(x), lambda greater than or equal to 0 and G(p) to be some appropriately defined functional space. They showed (g) over cap(tau),L(p) to be a linear spline for p=1 and parabolic spline for p=infinity, and suggested computations using conventional linear programming techniques. We describe a modification to the algorithm of Bartels and Conn (ACM Trans. Math. Software, 6 (1980)) for linearly constrained discrete L(1) problems and show how it can be utilized to compute the quantile smoothing splines. We also demonstrate how monotonicity and convexity constraints on the conditional quantile functions can be imposed easily. The parametric linear programming approach to computing all distinct tau th quantile smoothing splines for a given penalty parameter lambda, as well as all the quantile smoothing splines corresponding to all distinct lambda values for a given tau, also are provided.