An algorithm for quantile smoothing splines

Abstract For p = 1 and ∞, Koenker, Ng and Portnoy (Statistical Data Analysis Based on the L1 Norm and Related Methods (North-Holland, New York, 1992); Biometrika, 81 (1994)) proposed the τth Lp quantile smoothing spline, g τ ,L p , defined to solve min g∈ G p “fidelity” + λ “L p roughness” as a simple, nonparametric approach to estimating the τth conditional quantile functions given 0≤τ≤ 1. They defined “fidelity” = σi=1nϱτ(yi−g(xi)) with ϱτ(u)=(τ−I(u σ i=1 n−1 ¦g′(x i+1 )−g′(x i )¦ , “L∞ roughness” = maxxg″(x), λ⩾0 and G p to be some appropriately defined functional space. They showed g τ ,L p to be a linear spline for p=1 and parabolic spline for p=∞, 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 L1 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 τth quantile smoothing splines for a given penalty parameter λ, as well as all the quantile smoothing splines corresponding to all distinct λ values for a given τ, also are provided.