The Annals of Statistics

Block threshold rules for curve estimation using kernel and wavelet methods

Peter Hall, Gérard Kerkyacharian, and Dominique Picard

Full-text: Open access

Abstract

Motivated by recently developed threshold rules for wavelet estimators, we suggest threshold methods for general kernel density estimators, including those of classical Rosenblatt–Parzen type. Thresholding makes kernel methods competitive in terms of their adaptivity to a wide variety of aberrations in complex signals. It is argued that term-by-term thresholding does not always produce optimal performance, since individual coefficients cannot be estimated sufficiently accurately for reliable decisions to be made. Therefore, we suggest grouping coefficients into blocks and making simultaneous threshold decisions about all coefficients within a given block. It is argued that block thresholding has a number of advantages, including that it produces adaptive estimators which achieve minimax-optimal convergence rates without the logarithmic penalty that is sometimes associated with term-by-term thresholding. More than this, the convergence rates are achieved over large classes of functions with discontinuities, indeed with a number of discontinuities that diverges polynomially fast with sample size. These results are also established for block thresholded wavelet estimators, which, although they can be interpreted within the kernel framework, are often most conveniently constructed in a slightly different way.

Article information

Source
Ann. Statist., Volume 26, Number 3 (1998), 922-942.

Dates
First available in Project Euclid: 21 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1024691082

Digital Object Identifier
doi:10.1214/aos/1024691082

Mathematical Reviews number (MathSciNet)
MR1635418

Zentralblatt MATH identifier
0929.62040

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

Keywords
Adaptivity bias convergence rate density estimation minimax nonparametric regression smoothing parameter variance

Citation

Hall, Peter; Kerkyacharian, Gérard; Picard, Dominique. Block threshold rules for curve estimation using kernel and wavelet methods. Ann. Statist. 26 (1998), no. 3, 922--942. doi:10.1214/aos/1024691082. https://projecteuclid.org/euclid.aos/1024691082


Export citation

References

  • BERGH, J. and LOFSTROM, J. 1976. Interpolation Spaces: An Introduction. Springer, New York. ¨ ¨ Z.
  • DAUBECHIES, I. 1992. Ten Lectures on Wavelets. SIAM, Philadelphia. Z.
  • DEVORE, R. A. and LORENZ, G. G. 1993. Constructive Approximation. Springer, Berlin. Z.
  • DONOHO, D. and JOHNSTONE, I. M. 1996. Neoclassical minimax problems, thresholding and adaptive function estimation. Bernoulli 2 39 62. Z.
  • DONOHO, D., JOHNSTONE, I. M., KERKy ACHARIAN, G. and PICARD, D. 1993. Density estimation by wavelet thresholding. Technical Report 426, Dept. Statistics, Stanford Univ. Z.
  • DONOHO, D., JOHNSTONE, I. M., KERKy ACHARIAN, G. and PICARD, D. 1995. Wavelet shrinkage: Z. asy mptopia? with discussion. J. Roy. Statist. Soc. Ser. B 57 301 369. Z.
  • EFROIMOVITCH, S. Y. 1985. Nonparametric estimation of a density of unknown smoothness. Theory Probab. Appl. 30 557 661. Z.
  • HALL, P., KERKy ACHARIAN, G. and PICARD, D. 1995. On the minimax optimality of block thresholded wavelet estimators. Statist. Sinica. To appear. Z.
  • HALL, P. and PATIL, P. 1995. Formulae for mean integrated squared error of nonlinear waveletbased density estimators. Ann. Statist. 23 905 928.
  • HALL, P. and PATIL, P. 1996a. On the choice of smoothing parameter, threshold and truncation in nonparametric regression by nonlinear wavelet methods. J. Roy. Statist. Soc. Ser. B 58 361 377. Z.
  • HALL, P. and PATIL, P. 1996c. Effect of threshold rules on performance of wavelet-based curve estimators. Statist. Sinica 6 331 345. Z.
  • HALL, P., PENEV, S., KERKy ACHARIAN, G. and PICARD, D. 1997. Numerical performance of block thresholded wavelet estimators. Statist. Comput. 7 115 124. Z.
  • HARDLE, W., KERKy ACHARIAN, G., PICARD, D. and TSy BAKOV, A. B. 1996. Wavelets, Approxima¨ tion and Statistical Applications. Seminar Berlin, Paris. Z.
  • JOHNSTONE, I. M., KERKy ACHARIAN, G. and PICARD, D. 1992. Estimation d'une densite de ´ probabilite par methode d'ondelettes. C. R. Acad. Sci. Paris Ser. I Math. 315 ´ ´ 211 216. Z.
  • KERKy ACHARIAN, G. and PICARD, D. 1992. Density estimation in Besov spaces. Statist. Probab. Lett. 13 15 24. Z.
  • KERKy ACHARIAN, G. and PICARD, D. 1993. Density estimation by kernel and wavelet methods, optimality in Besov spaces. Statist. Probab. Lett. 18 327 336. Z.
  • KERKy ACHARIAN, G., PICARD, D. and TRIBOULEY, K. 1994. L adaptive density estimation. P Technical Report, Univ. Paris VII. Z.
  • LEPSKII, O. V. 1990. A problem of adaptive estimation in Gaussian white noise. Theory Probab. Appl. 35 454 466. Z.
  • LEPSKII, O. V. 1991. Asy mptotic minimax estimation I. Upper bounds. Theory Probab. Appl. 36 459 470. Z.
  • LEPSKII, O. V. 1992. Asy mptotic minimax estimation II. Models without optimal estimation, adaptive estimators. Theory Probab. Appl. 37 654 659. Z.
  • LEPSKII, O. V., MAMMEN E. and SPOKOINY, V. G. 1995. Adaptive spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimates with variable bandwidth selectors. Preprint. Institut fur Angewandte Analy sis und Stochastik, Berlin. ¨ Z.
  • LEPSKII, O. V. and SPOKOINY, V. G. 1995. Local adaptivity to inhomogeneous smoothness I. Resolution level. Preprint. Institut fur Angewandte Analy sis und Stochastik, Berlin. ¨ Z.
  • MEy ER, Y. 1990. Ondelettes. Hermann, Paris. Z.
  • PEETRE, J. 1975. New thoughts on Besov Spaces. Duke Univ. Press. Z.
  • TALAGRAND, M. 1994. Sharper bounds for empirical processes. Ann. Probab. 22 28 76. Z.
  • TRIEBEL, H. 1992. Theory of Function Spaces II. Birkhauser, Basel. ¨
  • CANBERRA, ACT 0200 33 RUE SAINT-LEU AUSTRALIA 80039 AMIENS, CEDEX 01 E-MAIL: halpstat@pretty.anu.edu.au FRANCE
  • 75251 PARIS, CEDEX 05 FRANCE