### 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 P_{c}(y_{i}_g{(x_{i})}+λ (int^{1}_{0}lg^{n}(x)/^{p}dx)^{1/p} with p_{t}(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 l_{1}-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 y_{i}'s. The entire path of solutions, in the quantile parameter τ, or the penalty parameter λ_{2}, 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 language | English (US) |
---|---|

Pages (from-to) | 673-680 |

Number of pages | 8 |

Journal | Biometrika |

Volume | 81 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1994 |

Externally published | Yes |

### Fingerprint

### 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

*Biometrika*,

*81*(4), 673-680. https://doi.org/10.1093/biomet/81.4.673

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

Research output: Contribution to journal › Article

*Biometrika*, vol. 81, no. 4, pp. 673-680. https://doi.org/10.1093/biomet/81.4.673

}

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 -