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Research Report SRR95-029
Performance of wavelet methods for functions with many discontinuities
Peter Hall, Ian McKay, Berwin A. Turlach
Abstract:
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. In the present paper, 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. Using
both theoretical and numerical methods 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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