QUANTILE SMOOTHING SPLINES

Although nonparametric regression has traditionally focused on the est imation of conditional mean functions, nonparametric estimation of con ditional quantile functions is often of substantial practical interest . We explore a class of quantile smoothing splines, defined as solutio ns to [GRAPHICS] with p(tau)(u)= u{tau - I(u < 0)}, p greater than or equal to 1, and appropriately chosen g. For the particular choices p = 1 and p = infinity we characterise solutions (g) over cap as splines, and discuss computation by standard l(1)-type linear programming tech niques. At lambda = 0, (g) over cap interpolates the tau th quantiles at the distinct design points, and for lambda sufficiently large (g) o ver cap is the linear regression quantile fit (Koenker & Bassett, 1978 ) to the observations. Because the methods estimate conditional quanti le functions they possess an inherent robustness to extreme observatio ns in the y(i)'s. The entire path of solutions, in the quantile parame ter tau, or the penalty parameter lambda, may be efficiently computed by parametric linear programming methods. We note that the approach ma y be easily adapted to impose monotonicity and/or convexity constraint s on the fitted function. An example is provided to illustrate the use of the proposed methods.