An algorithm for quantile smoothing splines

Pin T. Ng

Research output: Contribution to journalArticlepeer-review

29 Scopus citations


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, ĝτ,Lp, defined to solve min "fidelity" + λ "Lp roughness" g∈script G signp as a simple, nonparametric approach to estimating the τth conditional quantile functions given 0 ≤ τ ≤ 1. They defined "fidelity" = ∑ni-1 ρτi - g(xi)) with ρτ(u) = (τ - I(u < 0))u, "Li roughness" = ∑n-1i=1 |g′ (xi + 1) - g′ (xi)|, "L, roughness" = maxx g″ (x), λ ≥ 0 and script G signp to be some appropriately defined functional space. They showed ĝτ, Lp 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.

Original languageEnglish (US)
Pages (from-to)99-118
Number of pages20
JournalComputational Statistics and Data Analysis
Issue number2
StatePublished - Jul 1 1996


  • Constrained optimization
  • Monotone regression
  • Nonparametric regression
  • Quantile
  • Robustness
  • Splines

ASJC Scopus subject areas

  • Statistics and Probability
  • Computational Mathematics
  • Computational Theory and Mathematics
  • Applied Mathematics


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