Electronic Journal of Statistics

Data-driven wavelet-Fisz methodology for nonparametric function estimation

Piotr Fryzlewicz

Full-text: Open access

Abstract

We propose a wavelet-based technique for the nonparametric estimation of functions contaminated with noise whose mean and variance are linked via a possibly unknown variance function. Our method, termed the data-driven wavelet-Fisz technique, consists of estimating the variance function via a Nadaraya-Watson estimator, and then performing a wavelet thresholding procedure which uses the estimated variance function and local means of the data to set the thresholds at a suitable level.

We demonstrate the mean-square near-optimality of our wavelet estimator over the usual range of Besov classes. To achieve this, we establish an exponential inequality for the Nadaraya-Watson variance function estimator.

We discuss various implementation issues concerning our wavelet estimator, and demonstrate its good practical performance. We also show how it leads to a new wavelet-domain data-driven variance-stabilising transform. Our estimator can be applied to a variety of problems, including the estimation of volatilities, spectral densities and Poisson intensities, as well as to a range of problems in which the distribution of the noise is unknown.

Article information

Source
Electron. J. Statist., Volume 2 (2008), 863-896.

Dates
First available in Project Euclid: 1 October 2008

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1222868026

Digital Object Identifier
doi:10.1214/07-EJS139

Mathematical Reviews number (MathSciNet)
MR2447343

Zentralblatt MATH identifier
1320.62090

Subjects
Primary: 62G08: Nonparametric regression
Secondary: 62G05: Estimation 62G20: Asymptotic properties

Keywords
Besov spaces exponential inequality heteroscedasticity Nadaraya-Watson estimator nonparametric regression variance function variance-stabilising transform wavelets

Citation

Fryzlewicz, Piotr. Data-driven wavelet-Fisz methodology for nonparametric function estimation. Electron. J. Statist. 2 (2008), 863--896. doi:10.1214/07-EJS139. https://projecteuclid.org/euclid.ejs/1222868026


Export citation

References

  • F. Abramovich, Y. Benjamini, D. Donoho, and I. Johnstone. Adapting to unknown sparsity by controlling the false discovery rate., Ann. Statist., 34:584–653, 2006.
  • A. Antoniadis, P. Besbeas, and T. Sapatinas. Wavelet shrinkage for natural exponential families with cubic variance functions., Sankhya Ser. A, 63:309–327, 2001.
  • A. Antoniadis and T. Sapatinas. Wavelet shrinkage for natural exponential families with quadratic variance functions., Biometrika, 88:805–820, 2001.
  • K.J. Archer. Graphical technique for identifying a monotonic variance stabilizing transformation for absolute gene intensity signals., BMC Bioinformatics, 5:60, 2004.
  • P. Besbeas, I. De Feis, and T. Sapatinas. A comparative study of wavelet shrinkage estimators for Poisson counts., Int. Statist. Review, 72:209–237, 2004.
  • L. Breiman and J.H. Friedman. Estimating optimal transformations for multiple regression and correlation., J. Am. Statist. Ass., 80:580–619, 1985.
  • L.D. Brown, T. Cai, R. Zhang, L. Zhao, and H. Zhou. The root-unroot algorithm for density estimation as implemented via wavelet block thresholding., Preprint, 2007.
  • T. Cai and L. Wang. Adaptive variance function estimation in heteroscedastic nonparametric regression., Ann. Stat., to appear, 2007.
  • J-M. Chiou and H-G. Müller. Nonparametric quasi-likelihood., Ann. Stat., 27:36–64, 1999.
  • H. Dette and K. Pilz. On the estimation of a monotone conditional variance in nonparametric regression., Ann. Inst. Stat. Math., to appear, 2007.
  • D. Donoho and I. Johnstone. Adapting to unknown smoothness via wavelet shrinkage., J. Amer. Statist. Assoc., 90 :1200–1224, 1995.
  • D. L. Donoho and I. M. Johnstone. Ideal spatial adaptation by wavelet shrinkage., Biometrika, 81:425–455, 1994.
  • M. Fisz. The limiting distribution of a function of two independent random variables and its statistical application., Colloquium Mathematicum, 3:138–146, 1955.
  • P. Fryzlewicz, V. Delouille, and G. Nason. GOES-8 X-ray sensor variance stabilization using the multiscale data-driven Haar-Fisz transform., J. Roy. Statist. Soc. C, 56:99–116, 2007.
  • P. Fryzlewicz and G. P. Nason. A Haar-Fisz algorithm for Poisson intensity estimation., Journal of Computational and Graphical Statistics, 13:621–638, 2004.
  • P. Fryzlewicz, G.P. Nason, and R. von Sachs. A wavelet-Fisz approach to spectrum estimation., J. Time Ser. Anal., 29:868–880, 2008.
  • P. Fryzlewicz, T. Sapatinas, and S. Subba Rao. A Haar-Fisz technique for locally stationary volatility estimation., Biometrika, 93:687–704, 2006.
  • H-Y. Gao. Choice of thresholds for wavelet shrinkage estimate of the spectrum., J. Time Ser. Anal., 18:231–252, 1997a.
  • H-Y. Gao. Wavelet shrinkage estimates for heteroscedastic regression models. Statistical Sciences Division, MathSoft, Inc., 1997b.
  • T. Gasser, A. Kneip, and W. Koehler. A flexible and fast method for automatic smoothing., J. Amer. Statist. Assoc., 86:643–652, 1991.
  • M. Jansen. Multiscale Poisson data smoothing., J. R. Statist. Soc. B, 68:27–48, 2006.
  • M. Jansen, G.P. Nason, and B.W. Silverman. Multiscale methods for data on graphs and irregular multidimensional situations., J. R. Statist. Soc. Series B, to appear, 71, 2009.
  • W.B. Johnson, G. Schechtman, and J. Zinn. Best constants in moment inequalities for linear combinations of independent and exchangeable random variables., Ann. Prob., 13:234–253, 1985.
  • I. Johnstone and B. Silverman. Wavelet threshold estimators for data with correlated noise., J. Roy. Statist. Soc. Ser. B, 59:319–351, 1997.
  • I. Johnstone and B. Silverman. Empirical Bayes selection of wavelet thresholds., Ann. Statist, 33 :1700–1752, 2005a.
  • I.M. Johnstone and B.W. Silverman. EbayesThresh: R and S-Plus programs for Empirical Bayes thresholding., J. Statist. Software, 12.8:1–38, 2005b.
  • C. Joutard. Sharp large deviations in nonparametric estimation., J. Nonpar. Statist., 18:293–306, 2006.
  • E. Kolaczyk and R. Nowak. Multiscale likelihood analysis and complexity penalized estimation., Ann. Statist., 32:500–527, 2004.
  • E. D. Kolaczyk. Bayesian multiscale models for Poisson processes., Journal of the American Statistical Association, 94:920–933, 1999.
  • O.B. Linton, R. Chen, N.S. Wang, and W. Härdle. An analysis of transformations for additive nonparametric regression., J. Am. Statist. Ass., 92 :1512–1521, 1997.
  • D. Louani. Some large deviations limit theorems in conditional nonparametric statistics., Statistics, 33:171–196, 1999.
  • S. Mallat. A theory for multiresolution signal decomposition: the wavelet representation., IEEE Trans. Pattn Anal. Mach. Intell., 11:674–693, 1989.
  • Y. Meyer., Wavelets and Operators. Cambridge University Press, 1992.
  • E. Motakis, G. Nason, P. Fryzlewicz, and G. Rutter. Variance stabilization and normalization for one-color microarray data using a data-driven multiscale approach., Bioinformatics, 22 :2547–2553, 2006.
  • P. Moulin. Wavelet thresholding techniques for power spectrum estimation., IEEE Trans. Sig. Proc., 42 :3126–3136, 1994.
  • G. P. Nason and B. W. Silverman. The stationary wavelet transform and some statistical applications. In A. Antoniadis and G. Oppenheim, editors, Lecture Notes in Statistics, vol. 103, pages 281–300. Springer-Verlag, 1995.
  • M. H. Neumann. Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series., Journal of Time Series Analysis, 17:601–633, 1996.
  • M. Pensky, B. Vidakovic, and D. de Canditiis. Bayesian decision theoretic scale-adaptive estimation of a log-spectral density., Stat. Sinica, 17:635–666, 2007.
  • D.M. Rocke and B.P. Durbin. A model for measurement error for gene expression arrays., J. Comput. Biol., 8:557–569, 2001.
  • H.P. Rosenthal. On the subspaces of, lp (p>2) spanned by sequences of independent random variables. Israel J. Math., 8:273–303, 1970.
  • R. Rudzkis, L. Saulis, and V. Statulevicius. A general lemma on probabilities of large deviations., Lithuanian Math. J., 18:226–238, 1978.
  • S. Sardy, A. Antoniadis, and P. Tseng. Automatic smoothing with wavelets for a wide class of distributions., Journal of Computational and Graphical Statistics, 13:399–421, 2004.
  • R. Tibshirani. Estimating transformations for regression via additivity and variance stabilization., J. Am. Stat. Ass., 83:394–405, 1988.
  • K. E. Timmermann and R. D. Nowak. Multiscale modeling and estimation of Poisson processes with application to photon-limited imaging., IEEE Transactions on Information Theory, 45:846–862, 1999.
  • B. Vidakovic., Statistical Modeling by Wavelets. Wiley, New York, 1999.
  • R. von Sachs and B. MacGibbon. Non-parametric curve estimation by wavelet thresholding with locally stationary errors., Scand. J. Statist., 27:475–499, 2000.
  • L. Wang, L. Brown, T. Cai, and M. Levine. Effect of mean on variance function estimation in nonparametric regression., Ann. Stat., 36:646–664, 2008.