### Abstract

For p = 1 and ∞, 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 τth L_{p} quantile smoothing spline, ĝ_{τ,Lp}, defined to solve min "fidelity" + λ "L_{p} roughness" g∈script G sign_{p} as a simple, nonparametric approach to estimating the τth conditional quantile functions given 0 ≤ τ ≤ 1. They defined "fidelity" = ∑^{n}_{i-1} ρ_{τ}(γ_{i} - g(x_{i})) with ρ_{τ}(u) = (τ - I(u < 0))u, "L_{i} roughness" = ∑^{n-1}_{i=1} |g′ (x_{i + 1}) - g′ (x_{i})|, "L_{∞}, roughness" = max_{x} g″ (x), λ ≥ 0 and script G sign_{p} to be some appropriately defined functional space. They showed ĝ_{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 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 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 language | English (US) |
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Pages (from-to) | 99-118 |

Number of pages | 20 |

Journal | Computational Statistics and Data Analysis |

Volume | 22 |

Issue number | 2 |

DOIs | |

State | Published - Jul 1 1996 |

Externally published | Yes |

### Keywords

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

### ASJC Scopus subject areas

- Statistics and Probability
- Computational Mathematics
- Computational Theory and Mathematics
- Applied Mathematics