Double-smoothing for bias reduction in local linear regression

Local linear regression 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, with an asymptotic bias of order h²$h^2$ and variance of order (nh)^-1$(\mathit{nh}{)}^{-1}$ (h is 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. 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 h⁴$h^4$. The proposed estimator has better performance than local linear regression in situations with considerable bias effects; it also has less variability and more easily overcomes the sparse data problem than local cubic regression. At boundary points, the proposed estimator is comparable to local linear regression. Simulation studies are conducted and an ethanol example is used to compare the new approach with other competitive methods.