Local Linear Fitting and Improved Etimation near Peaks

We look into the problem of estimating a regression function that exhibits peaks, i.e. discontinuities in the first derivative. In the neighbourhood of such a discontinuity we investigate the behaviour of the local linear fit and find that it is not consistent for estimating the first derivative, and for estimating the function itself it loses in the rate of convergence. Indeed, near such points the local linear fit suffers significant finite sample bias. An objective quantity that indicates the presence of discontinuities in the first derivative is proposed. It is then used to construct an estimator that equals the classical local linear fit when possible, but switches to one of the one-sided estimators when appropriate to reduce bias at or near peaks. A simulation study, including comparisons with other methods, investigates the finitesample behaviour of the proposed estimator. We illustrate the use of the method on real data examples.