The Annals of Statistics

Performance of wavelet methods for functions with many discontinuities

Peter Hall, Ian McKay, and Berwin A. Turlach

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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.

Article information

Ann. Statist., Volume 24, Number 6 (1996), 2462-2476.

First available in Project Euclid: 16 September 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation
Secondary: 62G20: Asymptotic properties

Density estimation discontinuity mean integrated squared error jump nonparametric regression threshold wavelet


Hall, Peter; McKay, Ian; Turlach, Berwin A. Performance of wavelet methods for functions with many discontinuities. Ann. Statist. 24 (1996), no. 6, 2462--2476. doi:10.1214/aos/1032181162.

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  • Daubechies, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia.
  • Donoho, D. L. and Johnstone, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika 81 425-455.
  • Donoho, D. L. and Johnstone, I. M. (1995). Minimax estimation via wavelet shrinkage. Unpublished manuscript.
  • 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.
  • Donoho, D. L., Johnstone, I. M., Kerky acharian, G. and Picard, D. (1996). Density estimation by wavelet thresholding. Ann. Statist. 24 508-539.
  • Doukhan, P. (1988). Formes de Toeplitz associ´ees a une analyse multi-´echele. C. R. Acad. Sci. Paris S´er. I Math. 306 663-668.
  • Hall, P. and Patil, P. (1995). Formulae for mean integrated squared error of nonlinear waveletbased density estimators. Ann. Statist. 23 905-928.
  • 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.
  • Kerky acharian, G. and Picard, D. (1993). Introduction aux ondelettes et estimation de densit´e, 1: introduction aux ondelettes et a l'analyse multiresolution. Lecture notes, Univ. Paris XII.
  • Meyer, Y. (1990). Ondelettes. Hermann, Paris.
  • Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, Berlin.
  • Strang, G. (1989). Wavelets and dilation equations: a brief introduction. SIAM Rev. 31 614-627.
  • Strang, G. (1993). Wavelet transforms versus Fourier transforms. Bull. Amer. Math. Soc. 28 288-305.