Quantile smoothing splines

Roger Koenker, Pin T Ng, Stephen Portnoy

Research output: Contribution to journalArticle

325 Citations (Scopus)

Abstract

SUMMARY: Although nonparametric regression has traditionally focused on the estimation of conditional mean functions, nonparametric estimation of conditional quantile functions is often of substantial practical interest. We explore a class of quantile smoothing splines, defined as solutions to minσ Pc(yi_g{(xi)}+λ (int10lgn(x)/pdx)1/p with pt(u)=u{t_I(u< )}, pages; 1, and appropriately chosen G. For the particular choices p = 1 and p = ∞ we characterise solutions g as splines, and discuss computation by standard l1-type linear programming techniques. At λ =0, g interpolates the τ th quantiles at the distinct design points, and for λ sufficiently large g is the linear regression quantile fit (Koenker & Bassett, 1978) to the observations. Because the methods estimate conditional quantile functions they possess an inherent robustness to extreme observations in the yi's. The entire path of solutions, in the quantile parameter τ, or the penalty parameter λ2, may be efficiently computed by parametric linear programming methods. We note that the approach may be easily adapted to impose monotonicity and/or convexity constraints on the fitted function. An example is provided to illustrate the use of the proposed methods.

Original languageEnglish (US)
Pages (from-to)673-680
Number of pages8
JournalBiometrika
Volume81
Issue number4
DOIs
StatePublished - Dec 1994
Externally publishedYes

Fingerprint

Smoothing Splines
Quantile
Splines
Linear Programming
Conditional Quantiles
Quantile Function
linear programming
Parametric Linear Programming
Regression Quantiles
Linear programming
Nonparametric Regression
Nonparametric Estimation
Linear regression
Spline
Monotonicity
Penalty
Convexity
Linear Models
Extremes
Interpolate

Keywords

  • Bandwidth selection
  • Nonparametric regression
  • Quantile
  • Smoothing
  • Spline

ASJC Scopus subject areas

  • Statistics, Probability and Uncertainty
  • Applied Mathematics
  • Mathematics(all)
  • Statistics and Probability
  • Agricultural and Biological Sciences (miscellaneous)
  • Agricultural and Biological Sciences(all)

Cite this

Quantile smoothing splines. / Koenker, Roger; Ng, Pin T; Portnoy, Stephen.

In: Biometrika, Vol. 81, No. 4, 12.1994, p. 673-680.

Research output: Contribution to journalArticle

Koenker, R, Ng, PT & Portnoy, S 1994, 'Quantile smoothing splines', Biometrika, vol. 81, no. 4, pp. 673-680. https://doi.org/10.1093/biomet/81.4.673
Koenker, Roger ; Ng, Pin T ; Portnoy, Stephen. / Quantile smoothing splines. In: Biometrika. 1994 ; Vol. 81, No. 4. pp. 673-680.
@article{ea987f4f8ac048999b9e0e84d021c7b8,
title = "Quantile smoothing splines",
abstract = "SUMMARY: Although nonparametric regression has traditionally focused on the estimation of conditional mean functions, nonparametric estimation of conditional quantile functions is often of substantial practical interest. We explore a class of quantile smoothing splines, defined as solutions to minσ Pc(yi_g{(xi)}+λ (int10lgn(x)/pdx)1/p with pt(u)=u{t_I(u< )}, pages; 1, and appropriately chosen G. For the particular choices p = 1 and p = ∞ we characterise solutions g as splines, and discuss computation by standard l1-type linear programming techniques. At λ =0, g interpolates the τ th quantiles at the distinct design points, and for λ sufficiently large g is the linear regression quantile fit (Koenker & Bassett, 1978) to the observations. Because the methods estimate conditional quantile functions they possess an inherent robustness to extreme observations in the yi's. The entire path of solutions, in the quantile parameter τ, or the penalty parameter λ2, may be efficiently computed by parametric linear programming methods. We note that the approach may be easily adapted to impose monotonicity and/or convexity constraints on the fitted function. An example is provided to illustrate the use of the proposed methods.",
keywords = "Bandwidth selection, Nonparametric regression, Quantile, Smoothing, Spline",
author = "Roger Koenker and Ng, {Pin T} and Stephen Portnoy",
year = "1994",
month = "12",
doi = "10.1093/biomet/81.4.673",
language = "English (US)",
volume = "81",
pages = "673--680",
journal = "Biometrika",
issn = "0006-3444",
publisher = "Oxford University Press",
number = "4",

}

TY - JOUR

T1 - Quantile smoothing splines

AU - Koenker, Roger

AU - Ng, Pin T

AU - Portnoy, Stephen

PY - 1994/12

Y1 - 1994/12

N2 - SUMMARY: Although nonparametric regression has traditionally focused on the estimation of conditional mean functions, nonparametric estimation of conditional quantile functions is often of substantial practical interest. We explore a class of quantile smoothing splines, defined as solutions to minσ Pc(yi_g{(xi)}+λ (int10lgn(x)/pdx)1/p with pt(u)=u{t_I(u< )}, pages; 1, and appropriately chosen G. For the particular choices p = 1 and p = ∞ we characterise solutions g as splines, and discuss computation by standard l1-type linear programming techniques. At λ =0, g interpolates the τ th quantiles at the distinct design points, and for λ sufficiently large g is the linear regression quantile fit (Koenker & Bassett, 1978) to the observations. Because the methods estimate conditional quantile functions they possess an inherent robustness to extreme observations in the yi's. The entire path of solutions, in the quantile parameter τ, or the penalty parameter λ2, may be efficiently computed by parametric linear programming methods. We note that the approach may be easily adapted to impose monotonicity and/or convexity constraints on the fitted function. An example is provided to illustrate the use of the proposed methods.

AB - SUMMARY: Although nonparametric regression has traditionally focused on the estimation of conditional mean functions, nonparametric estimation of conditional quantile functions is often of substantial practical interest. We explore a class of quantile smoothing splines, defined as solutions to minσ Pc(yi_g{(xi)}+λ (int10lgn(x)/pdx)1/p with pt(u)=u{t_I(u< )}, pages; 1, and appropriately chosen G. For the particular choices p = 1 and p = ∞ we characterise solutions g as splines, and discuss computation by standard l1-type linear programming techniques. At λ =0, g interpolates the τ th quantiles at the distinct design points, and for λ sufficiently large g is the linear regression quantile fit (Koenker & Bassett, 1978) to the observations. Because the methods estimate conditional quantile functions they possess an inherent robustness to extreme observations in the yi's. The entire path of solutions, in the quantile parameter τ, or the penalty parameter λ2, may be efficiently computed by parametric linear programming methods. We note that the approach may be easily adapted to impose monotonicity and/or convexity constraints on the fitted function. An example is provided to illustrate the use of the proposed methods.

KW - Bandwidth selection

KW - Nonparametric regression

KW - Quantile

KW - Smoothing

KW - Spline

UR - http://www.scopus.com/inward/record.url?scp=0001652263&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001652263&partnerID=8YFLogxK

U2 - 10.1093/biomet/81.4.673

DO - 10.1093/biomet/81.4.673

M3 - Article

VL - 81

SP - 673

EP - 680

JO - Biometrika

JF - Biometrika

SN - 0006-3444

IS - 4

ER -