Performance of wavelet methods for functions with many discontinuities.

*(English)*Zbl 0867.62029Summary: 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.

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.

##### MSC:

62G07 | Density estimation |

62G20 | Asymptotic properties of nonparametric inference |

42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |

##### Keywords:

density estimation; mean integrated squared error; jump; nonparametric regression; threshold; curve estimation; discontinuities; wavelet estimator; wavelet methods; rates of convergence; mean square consistency
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\textit{P. Hall} et al., Ann. Stat. 24, No. 6, 2462--2476 (1996; Zbl 0867.62029)

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