We consider the problem of estimating jump points in smooth curves. Observa-.., 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 Q ĉ(t) is constructed. Our main result is the convergence to a compound Poisson process with drift, of a local dilated-rescaled version of ĉ(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 h n . 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.
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