QUANTILE SMOOTHING SPLINES

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 [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 techniques. At lambda = 0, (g) over cap interpolates the tau th quantiles at the distinct design points, and for lambda sufficiently large (g) over cap 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 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.