(Received May 15, 1997; revised November 5, 1997)
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+delta (for some delta > 0) in the context of recent one-step methods based purely on kernels, or n-1(log n)1+delta 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.
Key words and phrases: Bandwidth, curve estimation, change point, diagnostic, discontinuity, kernel, least squares, nonparametric regression.
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