Performance of wavelet methods for functions with many discontinuities. (English) Zbl 0867.62029

Summary: 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.


62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
Full Text: DOI


[1] Daubechies, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia. · Zbl 0776.42018
[2] Donoho, D. L. and Johnstone, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika 81 425-455. JSTOR: · Zbl 0815.62019 · doi:10.1093/biomet/81.3.425
[3] Donoho, D. L. and Johnstone, I. M. (1995). Minimax estimation via wavelet shrinkage. Unpublished manuscript. · Zbl 0935.62041
[4] Donoho, D. L., Johnstone, I. M., Kerky acharian, G. and Picard, D. (1995). Wavelet shrinkage: asy mptopia? (with discussion). J. Roy. Statist. Soc. Ser. B 57 301-369. JSTOR: · Zbl 0827.62035
[5] Donoho, D. L., Johnstone, I. M., Kerky acharian, G. and Picard, D. (1996). Density estimation by wavelet thresholding. Ann. Statist. 24 508-539. · Zbl 0860.62032 · doi:10.1214/aos/1032894451
[6] Doukhan, P. (1988). Formes de Toeplitz associées a une analyse multi-échele. C. R. Acad. Sci. Paris Sér. I Math. 306 663-668. · Zbl 0661.42019
[7] Hall, P. and Patil, P. (1995). Formulae for mean integrated squared error of nonlinear waveletbased density estimators. Ann. Statist. 23 905-928. · Zbl 0839.62042 · doi:10.1214/aos/1176324628
[8] Hall, P., McKay, I. and Turlach, B. (1995). Performance of wavelet methods for functions with many discontinuities. Technical Report SRR 029-95, Centre for Mathematics and its Applications, Australian National Univ. · Zbl 0867.62029 · doi:10.1214/aos/1032181162
[9] Kerky acharian, G. and Picard, D. (1993). Introduction aux ondelettes et estimation de densité, 1: introduction aux ondelettes et a l’analyse multiresolution. Lecture notes, Univ. Paris XII.
[10] Meyer, Y. (1990). Ondelettes. Hermann, Paris. · Zbl 0735.42017
[11] Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, Berlin. · Zbl 0322.60043
[12] Strang, G. (1989). Wavelets and dilation equations: a brief introduction. SIAM Rev. 31 614-627. JSTOR: · Zbl 0683.42030 · doi:10.1137/1031128
[13] Strang, G. (1993). Wavelet transforms versus Fourier transforms. Bull. Amer. Math. Soc. 28 288-305. · Zbl 0771.42021 · doi:10.1090/S0273-0979-1993-00390-2
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.