### 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) |
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Pages (from-to) | 673-680 |

Number of pages | 8 |

Journal | Biometrika |

Volume | 81 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1994 |

Externally published | Yes |

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### 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