Performance of wavelet methods for functions with many discontinuities

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.