ON THE ESTIMATION OF JUMP POINTS IN SMOOTH CURVES

IRENE GIJBELS^{1}, PETER HALL^{1,2} AND ALOÏS KNEIP^{1}*

1Institut de Statistique, Université Catholique de Louvain, 20 Voie du Roman Pays,

B-1348 Louvain-la-Neuve, Belgium

2Centre for Mathematics and Its Applications, Australian National University, Canberra, ACT 0200, Australia

(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+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.

Key words and phrases: Bandwidth, curve estimation, change point, diagnostic, discontinuity, kernel, least squares, nonparametric regression.