The Annals of Statistics

Adaptive variance function estimation in heteroscedastic nonparametric regression

T. Tony Cai and Lie Wang

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We consider a wavelet thresholding approach to adaptive variance function estimation in heteroscedastic nonparametric regression. A data-driven estimator is constructed by applying wavelet thresholding to the squared first-order differences of the observations. We show that the variance function estimator is nearly optimally adaptive to the smoothness of both the mean and variance functions. The estimator is shown to achieve the optimal adaptive rate of convergence under the pointwise squared error simultaneously over a range of smoothness classes. The estimator is also adaptively within a logarithmic factor of the minimax risk under the global mean integrated squared error over a collection of spatially inhomogeneous function classes. Numerical implementation and simulation results are also discussed.

Article information

Ann. Statist. Volume 36, Number 5 (2008), 2025-2054.

First available in Project Euclid: 13 October 2008

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Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression 62G20: Asymptotic properties

Adaptive estimation nonparametric regression thresholding variance function estimation wavelets


Cai, T. Tony; Wang, Lie. Adaptive variance function estimation in heteroscedastic nonparametric regression. The Annals of Statistics 36 (2008), no. 5, 2025--2054. doi:10.1214/07-AOS509.

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  • Berger, J. (1976). Minimax estimation of a multivariate normal mean under arbitrary quadratic loss. J. Multivariate Anal. 4 642–648.
  • Brown, L. D. and Levine, M. (2007). Variance estimation in nonparametric regression via the difference sequence method. Ann. Statist. 35 2219–2232.
  • Brown, L. D. and Low, M. G. (1996). A constrained risk inequality with applications to nonparametric functional estimations. Ann. Statist. 24 2524–2535.
  • Cai, T. (1999). Adaptive wavelet estimation: A block thresholding and oracle inequality approach. Ann. Statist. 27 898–924.
  • Cai, T. (2002). On block thresholding in wavelet regression: Adaptivity, block size, and threshold level. Statist. Sinica 12 1241–1273.
  • Cai, T. and Silverman, B. W. (2001). Incorporating information on neighboring coefficients into wavelet estimation. Sankhyā Ser. B 63 127–148.
  • Cai, T. and Wang, L. (2007). Adaptive variance function estimation in heteroscedastic nonparametric regression. Technical report.
  • Daubechies, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia.
  • Donoho, D. L. and Johnstone, I. M. (1994). Ideal spatial adaptation via wavelet shrinkage. Biometrika 81 425–455.
  • Donoho, D. L., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1995). Wavelet shrinkage: Asymptopia? J. Roy. Statist. Soc. Ser. B 57 301–369.
  • Fan, J. and Yao, Q. (1998). Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85 645–660.
  • Hall, P. and Carroll, R. J. (1989). Variance function estimation in regression: The effect of estimating the mean. J. Roy. Statist. Soc. Ser. B 51 3–14.
  • Hall, P., Kerkyacharian, G. and Picard, D. (1998). Block threshold rules for curve estimation using kernel and wavelet methods. Ann. Statist. 26 922–942.
  • Hall, P., Kerkyacharian, G. and Picard, D. (1999). On the minimax optimality of block thresholded wavelet estimators. Statist. Sinica 9 33–50.
  • Hwang, H. K. (1996). Large deviations for combinatorial distributions. I. Central limit theorems. Ann. Appl. Probab. 6 297–319.
  • Johnstone, I. M. and Silverman, B. W. (2005). Empirical Bayes selection of wavelet thresholds. Ann. Statist. 33 1700–1752.
  • Lepski, O. V. (1990). On a problem of adaptive estimation on white Gaussian noise. Theory Probab. Appl. 35 454–466.
  • Meyer, Y. (1992). Wavelets and Operators. Cambridge Univ. Press.
  • Müller, H. G. and Stadtmüller, U. (1987). Estimation of heteroscedasticity in regression analysis. Ann. Statist. 15 610–625.
  • Müller, H. G. and Stadtmüller, U. (1993). On variance function estimation with quadratic forms. J. Statist. Plann. Inference 35 213–231.
  • Ruppert, D., Wand, M. P., Holst, U. and Hössjer, O. (1997). Local polynomial variance function estimation. Technometrics 39 262–273.
  • Triebel, H. (1983). Theory of Function Spaces. Birkhäuser, Basel.
  • Wang, L., Brown, L. D., Cai, T. and Levine, M. (2008). Effect of mean on variance function estimation on nonparametric regression. Ann. Statist. 36 646–664.