Open Access
December 1996 Performance of wavelet methods for functions with many discontinuities
Peter Hall, Ian McKay, Berwin A. Turlach
Ann. Statist. 24(6): 2462-2476 (December 1996). DOI: 10.1214/aos/1032181162


Compared to traditional approaches to curve estimation, such as those based on kernels, wavelet methods are relatively unaffected by discontinuities and similar aberrations. In particular, the mean square convergence rate of a wavelet estimator of a fixed, piecewise-smooth curve is not influenced by discontinuities. Nevertheless, it is clear that as the estimation problem becomes more complex the limitations of wavelet methods must eventually be apparent. By allowing the number of discontinuities to increase and their size to decrease as the sample grows, we study the limits to which wavelet methods can be pushed and still exhibit good performance. We determine the effect of these changes on rates of convergence. For example, we derive necessary and sufficient conditions for wavelet methods applied to increasingly complex, discontinuous functions to achieve convergence rates normally associated only with fixed, smooth functions, and we determine necessary conditions for mean square consistency.


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Peter Hall. Ian McKay. Berwin A. Turlach. "Performance of wavelet methods for functions with many discontinuities." Ann. Statist. 24 (6) 2462 - 2476, December 1996.


Published: December 1996
First available in Project Euclid: 16 September 2002

zbMATH: 0867.62029
MathSciNet: MR1425961
Digital Object Identifier: 10.1214/aos/1032181162

Primary: 62G07
Secondary: 62G20

Keywords: Density estimation , discontinuity , jump , mean integrated squared error , Nonparametric regression , threshold , ‎wavelet

Rights: Copyright © 1996 Institute of Mathematical Statistics

Vol.24 • No. 6 • December 1996
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