Statistica Sinica 9(1999), 759-774 SMOOTHING REGRESSION QUANTILE BY COMBINING k-NN ESTIMATION WITH LOCAL LINEAR KERNEL FITTING Keming Yu Lancaster University Abstract: A two-step nonparametric regression quantile smoothing technique is presented here, combining a standard k-NN technique and a locally linear kernel smoother. There are many advantages to this approach: an asymptotically optimal mean square error (Fan, Hu and Truong (1995)), a ready-made bandwidth selection rule (Yu and Jones (1998)), and simple computation and flexible estimation under variable transformations and distributional assumptions. The method is tested on a simulated example, and applied to data. Key words and phrases: Bandwidth selection, correlated regression model, double kernel method, k-NN method, local linear kernel smoothing, mean square error, regression quantile. 1. Introduction Quantile regression is widely used for screening some biometric measure- ments (height, weight, circumferences and skinfold) against an appropriate co- variate (age, time) (Healy, Rasbash and Yang (1988), Cole (1988), Goldstein and Pan (1992), Royston and Altman (1994)). Some extreme (high or low) quantiles of underlying distributions of measurements are particularly useful for indus- trial applications (Magee, Burbidge and Robb (1991), Hendricks and Koenker (1992)). In this area many advances in theory and application have been made in the last few years, and some nonparametric and semi-parametric techniques (Jones and Hall (1990), Bhattacharya and Gangopadhyay (1990), Cole and Green (1992), Fan, Hu and Truong (1995), Yu and Jones (1998)) are particularly at- tractive. Jones and Hall’s theoretical investigation based on the kernel fitting of the “check-function” ρ p (t)= {|t| + (2p - 1)t}/2 of Koenker and Bassett (1978) was extended by Fan, Hu and Truong with an advanced locally linear smoother. The asymptotic mean square error (AMSE) for internal and boundary points is given by AMSE(ˆ q p (x)) = μ 2 (K) 2 (q ′′ p (x)) 2 h 4 + R(K) nhg(x) , where ˆ q p (x) is the estimator of the true pth (0 <p< 1) quantile function q p (x) of response Y given covariate X = x, K is a symmetric function with