# Quantile Smoothing Splines

Roger Koenker, Pin Ng and Stephen Portnoy
Biometrika
Vol. 81, No. 4 (Dec., 1994), pp. 673-680
DOI: 10.2307/2337070
Stable URL: http://www.jstor.org/stable/2337070
Page Count: 8

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## Abstract

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 ming∈G ∑ ρτ {yi - g(xi)} + λ (∫1 0 |g''(x)|p dx)1/p, with $\rho_\tau(u) = u\{\tau - I(u < 0)\}, p \geqslant 1$, and appropriately chosen G. For the particular choices p = 1 and p = ∞ we characterise solutions ĝ as splines, and discuss computation by standard l1-type linear programming techniques. At λ = 0, ĝ interpolates the τth quantiles at the distinct design points, and for λ sufficiently large ĝ 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 λ, 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.

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