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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 $\min_{g\in {\mathcal G}}\sum \rho_ \tau \{y_ i- g(x_ i)\}+ \lambda\Biggl( \int_ 0^ 1 | g''(x) |^ p dx \Biggr)^{1/p},$ with $$\rho_ \tau(u)= u\{\tau- I(u< 0)\}$$, $$p\geq 1$$, and appropriately chosen $${\mathcal G}$$. For the particular choices $$p=1$$ and $$p=\infty$$ we characterise solutions $$\widehat{g}$$ as splines, and discuss computation by standard $$l_ 1$$-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 $$y_ i$$’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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