Change Point Estimation by Local Linear Smoothing

We consider the problem of estimating jump points in smooth curves. Observations (Xi,Yi) i=1,…,n from a random design regression function are given. We focus essentially on the basic situation where a unique change point is present in the regression function. Based on local linear regression, a jump estimate process t→γ(t) is constructed. Our main result is the convergence to a compound Poisson process with drift, of a local dilated-rescaled version of γ(t), under a positivity condition regarding the asymmetric kernel involved. This result enables us to prove that our estimate of the jump location converges with exact rate n−1 without any particular assumption regarding the bandwidth hn. Other consequences such as asymptotic normality are investigated and some proposals are provided for an extension of this work to more general situations. Finally we present Monte-Carlo simulations which give evidence for good numerical performance of our procedure.