The Annals of Statistics

Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra

Michael H. Neumann and Rainer von Sachs

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We derive minimax rates for estimation in anisotropic smoothness classes. These rates are attained by a coordinatewise thresholded wavelet estimator based on a tensor product basis with separate scale parameter for every dimension. It is shown that this basis is superior to its one-scale multiresolution analog, if different degrees of smoothness in different directions are present.

As an important application we introduce a new adaptive waveletestimator of the time-dependent spectrum of a locally stationary time series. Using this model which was recently developed by Dahlhaus, we show that the resulting estimator attains nearly the rate, which is optimal in Gaussian white noise, simultaneously over a wide range of smoothness classes. Moreover, by our new approach we overcome the difficulty of how to choose the right amount of smoothing, that is, how to adapt to the appropriate resolution, for reconstructing the local structure of the evolutionary spectrum in the time-frequency plane.

Article information

Ann. Statist., Volume 25, Number 1 (1997), 38-76.

First available in Project Euclid: 10 October 2002

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Zentralblatt MATH identifier

Primary: 62G07: Density estimation 62M15: Spectral analysis
Secondary: 62E20: Asymptotic distribution theory 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Anisotropic smoothness classes adaptive estimation optimal rate of convergence wavelet thresholding tensor product basis time-frequency plane locally stationary time series evolutionary spectrum


Neumann, Michael H.; von Sachs, Rainer. Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra. Ann. Statist. 25 (1997), no. 1, 38--76. doi:10.1214/aos/1034276621.

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  • BERKOLAJKO, M. Z. and NOVIKOV, I. YA. 1992. Bases of wavelets in spaces of differentiable functions of anisotropic smoothness. Russian Acad. Sci. Dokl. Math. 45 382 386. Z.
  • BEy LKIN, G., COIFMAN, R. and ROKHLIN, V. 1993. Fast wavelet transforms and numerical algorithms I. Comm. Pure Appl. Math. 44 141 183. Z.
  • BRETAGNOLLE, J. and HUBER, C. 1979. Estimation des densities: risque minimax. Z. Wahrsch. ´ Verw. Gebiete 47 119 137. Z.
  • BROWN, D. L. and LOW, M. 1996. Asy mptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384 2398. Z.
  • COHEN, A., DAUBECHIES, I. and VIAL, P. 1993. Wavelets on the interval and fast wavelet transform. Appl. Comput. Harmonic Anal. 1 54 81. Z.
  • DAHLHAUS, R. 1996a. On the Kullback Leibler information divergence of locally stationary processes. Stochastic Process. Appl. 62 139 168. Z.
  • DAHLHAUS, R. 1996b. Asy mptotic statistical inference for nonstationary processes with evolutionary spectra. In Athens Conference on Applied Probability and Time Series Analy Z. sis P. M. Robinson and M. Rosenblatt, eds. 2. Springer, New York. Z.
  • DAHLHAUS, R. 1997. Fitting time series models to nonstationary processes. Ann. Statist. 25 1 37.Z.
  • DAUBECHIES, I. 1992. Ten Lectures on Wavelets. SIAM, Philadelphia. Z.
  • DELy ON, B. and JUDITSKY, A. 1996. On minimax wavelet estimators. Appl. Comput. Harmonic Anal. 3 215 228. Z.
  • DONOHO, D. L. 1995. CART and best-ortho-basis: a connection. Technical report, Dept. Statistics, Stanford Univ. Z.
  • DONOHO, D. L. and JOHNSTONE, I. M. 1994a. Ideal spatial adaptation by wavelet shrinkage. Biometrika 81 425 455. Z.
  • DONOHO, D. L. and JOHNSTONE, I. M. 1994b. Ideal denoising in an orthonormal basis chosen from a library of bases. C. R. Acad. Sci. Paris Ser. I Math. 319 1317 1322. ´ Z.
  • DONOHO, D. L., 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.
  • EDWARDS, R. E. 1979. Fourier Series. A Modern Introduction 1, 2nd ed. Springer, New York. Z.
  • GAO, H.-Y. 1993. Wavelet estimation of spectral densities in time series analysis. Ph.D. dissertation, Univ. California, Berkeley. Z.
  • MARTIN, W. and FLANDRIN, P. 1985. Wigner Ville spectral analysis of nonstationary processes. IEEE Trans. Signal Process. 33 1461 1470. Z.
  • MEy ER, Y. 1991. Ondelettes sur l'intervalle. Rev. Math. Iberoamericana 7 115 133. Z.
  • NEUMANN, M. H. 1994. Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series. J. Time Ser. Anal. To appear. Z.
  • NEUMANN, M. H. and SPOKOINY, V. G. 1995. On the efficiency of wavelet estimators under arbitrary error distributions. Math. Methods Statist. 4 137 166.
  • NEUMANN, M. H. and VON SACHS, R. 1995. Wavelet thresholding: bey ond the Gaussian i.i.d. situation. Wavelets and Statistics. Lecture Notes in Statist. 301 329. Springer, New York. Z. NIKOL'SKII, S. M. 1975. Approximation of Functions of Several Variables and Imbedding Theorems. Springer, Berlin. Z.
  • NUSSBAUM, M. 1996. Asy mptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 2399 2430. Z.
  • PRIESTLEY, M. B. 1981. Spectral Analy sis and Time Series 2. Academic Press, London. Z.
  • RUDZKIS, R., SAULIS, L. and STATULEVICIUS, V. 1978. A general lemma on probabilities of large deviations. Lithuanian Math. J. 18 226 238. Z.
  • TRIBOULEY, K. 1995. Practical estimation of multivariate densities using wavelet methods. Statist. Neerlandica 49 41 62. Z.
  • VON SACHS, R. and SCHNEIDER, K. 1996. Wavelet smoothing of evolutionary spectra by nonlinear thresholding. Appl. Comput. Harmonic Anal. 3 268 282. Z.
  • WALSH, J. B. 1986. Martingales with a multidimensional parameter and stochastic integrals in the plane. Lectures in Probability and Statistics. Lecture Notes in Math. 1215 329 491. Springer, Berlin.