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Quantile smoothing splines. (English) Zbl 0810.62040
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 $$\min\sb{g\in {\cal G}}\sum \rho\sb \tau \{y\sb i- g(x\sb i)\}+ \lambda\Biggl( \int\sb 0\sp 1 \vert g''(x) \vert\sp p dx \Biggr)\sp{1/p},$$ with $\rho\sb \tau(u)= u\{\tau- I(u< 0)\}$, $p\geq 1$, and appropriately chosen ${\cal G}$. For the particular choices $p=1$ and $p=\infty$ we characterise solutions $\widehat{g}$ as splines, and discuss computation by standard $l\sb 1$-type linear programming techniques. At $\lambda=0$, $\widehat{g}$ interpolates the $\tau$th quantiles at the distinct design points, and for $\lambda$ sufficiently large $\widehat{g}$ is the linear regression quantile fit to the observations. Because the methods estimate conditional quantile functions they possess an inherent robustness to extreme observations in the $y\sb i$’s. The entire path of solutions, in the quantile parameter $\tau$, or the penalty parameter $\lambda$, 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.

##### MSC:
 62G07 Density estimation
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