Double-smoothing for Bias Reduction in Local Linear Regression

Local linear regression is commonly used in practice because of its excellent numerical and theoretical properties. It involves fitting a straight line segment over a small region whose midpoint is the target point x, and the local linear estimate at x is the estimated intercept of that straight line segment. Local linear estimator has an asymptotic bias of order h and variance of order (nh) with h the bandwidth. In this paper, we propose a new estimator, the double-smoothing local linear estimator, which is constructed by integrally combining all fitted values at x of local lines in its neighborhood with another round of smoothing. In contrast to using only an intercept in the conventional local linear regression, the proposed estimator attempts to make use of all information obtained from fitting local lines. Without changing the order of variance, the new estimator can reduce the bias to an order of h. Compared to local linear regression, the proposed estimator has better performance in situations with considerable bias effects; compared to local cubic regression, with the same convergence rate for the asymptotic bias and variance, the new estimator has less variability and more easily overcomes the sparse data problem because the design matrix is used only in the first step of smoothing. At boundary points, the proposed estimator is comparable to local linear regression. Simulation results and a real data