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ON THE ESTIMATION OF JUMP POINTS

IN SMOOTH CURVES

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

^{1} *Institut de Statistique, Université Catholique de Louvain, 20 Voie du Roman Pays,*

B-1348 Louvain-la-Neuve, Belgium

^{2} *Centre 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+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.

**Source**
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