Abstract
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.
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Gijbels, I., Hall, P. & Kneip, A. On the Estimation of Jump Points in Smooth Curves. Annals of the Institute of Statistical Mathematics 51, 231–251 (1999). https://doi.org/10.1023/A:1003802007064
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DOI: https://doi.org/10.1023/A:1003802007064