### 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) |
---|---|

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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Statistics, Probability and Uncertainty
- Electrical and Electronic Engineering
- Computational Mathematics
- Numerical Analysis
- Statistics and Probability

### Cite this

**An algorithm for quantile smoothing splines.** / Ng, Pin T.

Research output: Contribution to journal › Article

*Computational Statistics and Data Analysis*, vol. 22, no. 2, pp. 99-118. https://doi.org/10.1016/0167-9473(95)00044-5

}

TY - JOUR

T1 - An algorithm for quantile smoothing splines

AU - Ng, Pin T

PY - 1996/7/1

Y1 - 1996/7/1

N2 - 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 ρτ(γi - 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.

AB - 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 ρτ(γi - 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.

KW - Constrained optimization

KW - Monotone regression

KW - Nonparametric regression

KW - Quantile

KW - Robustness

KW - Splines

UR - http://www.scopus.com/inward/record.url?scp=0030192103&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030192103&partnerID=8YFLogxK

U2 - 10.1016/0167-9473(95)00044-5

DO - 10.1016/0167-9473(95)00044-5

M3 - Article

VL - 22

SP - 99

EP - 118

JO - Computational Statistics and Data Analysis

JF - Computational Statistics and Data Analysis

SN - 0167-9473

IS - 2

ER -