Quantile smoothing splines

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 minσ P_c(y_i_g{(x_i)}+λ (int¹₀lgⁿ(x)/^pdx)^1/p with p_t(u)=u{t_I(u< )}, pages; 1, and appropriately chosen G. For the particular choices p = 1 and p = ∞ we characterise solutions g as splines, and discuss computation by standard l₁-type linear programming techniques. At λ =0, g interpolates the τ th quantiles at the distinct design points, and for λ sufficiently large g 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 τ, or the penalty parameter λ₂, 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.