On the estimation of jump points in smooth curves

Two-step methods are suggested for obtaining optimal performance in the problem of estimating jump points in smooth curves. The first step is based on a kernel-type diagnostic, and the second on local least-squares. In the case of a sample of size n the exact convergence rate is n^-1, rather than n^-1+δ (for some δ > 0) in the context of recent one-step methods based purely on kernels, or n^-1(log n)^1+δ for recent techniques based on wavelets. Relatively mild assumptions are required of the error distribution. Under more stringent conditions the kernel-based step in our algorithm may be used by itself to produce an estimator with exact convergence rate n^-1(log n)1/2. Our techniques also enjoy good numerical performance, even in complex settings, and so offer a viable practical alternative to existing techniques, as well as providing theoretical optimality.