On Bias Reduction in Local Linear Smoothing

The standard approach to local linear regression involves fitting a straight line segment to a curve in a symmetrical way, in that the segment is fitted directly above a small region whose midpoint is the abscissa, x, at which we wish to estimate the curve. In this paper we show that, if the segment is fitted in a skew manner, with its centre a little to the left or right of x, then bias can be reduced by an order of magnitude, without affecting the order of magnitude of variance. The amount by which the centre should be shifted depends only on the kernel function, and not at all on the unknown regression mean or on the design density. The average of two similarly but oppositely shifted estimators has two orders of magnitude less bias, again at the expense of a slight increase in variance. This particular estimator may be viewed as a limiting form of a convex combination of three local linear estimators, the two oppositely shifted estimators and the symmetric one, which has two orders of magnitude less bias and, depending on the kernel function, less variance as well.