Skip to main content
SpringerLink
Log in

Nonparametric Empirical Bayes Estimation via Wavelets

  • Chapter
Bayesian Inference in Wavelet-Based Models

Part of the book series: Lecture Notes in Statistics ((LNS,volume 141))

Abstract

We consider a traditional nonparametric empirical Bayes (EB) setting with a parameter θ being a location or a scale parameter. EB estimators are constructed and are shown to provide adaptation to the unknown degree of smoothness of g(θ). The case when the conditional distributions belongs to a one-parameter exponential family is also studied. Examples are given for familiar families of conditional distributions. The advantages of using wavelets are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Antoniadies, A., Grégoire, G., and McKeague, I.W. (1994) Wavelet method for curve estimation. J. Amer. Statist. Assoc, 89, 1340–1353.

    Article  MathSciNet  Google Scholar 

  • Brown, L.D. and Low M.G. (1996) A constrained risk inequality with applications to nonparametric functional estimation. Ann. Stat., 24, 2524–2535.

    Article  MathSciNet  MATH  Google Scholar 

  • Carlin, B.P. and Louis, T.A. (1996) Bayes and Empirical Bayes Methods for Data Analysis, Chapman & Hall, London.

    MATH  Google Scholar 

  • Daubechies, I. (1992) Ten Lectures in Wavelets, SIAM, Philadelphia.

    Book  Google Scholar 

  • Donoho, D.L. (1994) Smooth wavelet decompositions with blocky coefficient kernels. In: Recent Advances in Wavelet Analysis, ed. by Schumaker, L.L. and Webb, G., Academic Press, London, 259–308.

    Google Scholar 

  • Donoho, D.L. and Johnstone, I.M. (1995) Adapting to unknown smoothness via wavelet shrinkage. J. Amer. Statist. Assoc, 90, 1200–1224.

    Article  MathSciNet  MATH  Google Scholar 

  • Donoho, D.L., Johnstone, I.M., Kerkyacharian, G., and Picard, D. (1996) Density estimation by wavelet thresholding. Ann. Stat., 24, 508–539.

    Article  MathSciNet  MATH  Google Scholar 

  • Erdélyi, A. et al (1954) Tables of Integral Transforms. Mc Graw-Hill Book Company, New York.

    Google Scholar 

  • Hall, P., and Patil, P. (1995) Formulae for mean integrated squared error of nonlinear wavelet-based density estimators. Ann. Stat., 23, 905–928.

    Article  MathSciNet  MATH  Google Scholar 

  • Hall, P., Kerkyacharian, G., and Picard, D. (1998) Block thresholding rules for curve estimation using kernel and wavelet methods. Ann. Stat, 26, 922–942.

    Article  MathSciNet  MATH  Google Scholar 

  • Hernández, E., and Weiss, G. (1996) A First Course on Wavelets, CRC Press, Boca Raton.

    Book  MATH  Google Scholar 

  • Huang, Su-Yun (1997) Wavelet based empirical Bayes estimation for the uniform distribution. Statist. Probab. Lett., 32, 141–146.

    Article  MathSciNet  MATH  Google Scholar 

  • Kerkyacharian, G., and Picard, D. (1992) Density estimation in Besov spaces. Statist. Probab. Lett, 13, 15–24.

    Article  MathSciNet  MATH  Google Scholar 

  • Lahiri, P., and Park, D.H. (1991) Nonparametric Bayes and empirical Bayes estimators of mean residual life at age t, J. Stat. Plan. Inference, 29, 125–136.

    Article  MathSciNet  MATH  Google Scholar 

  • Maritz, J., and Lwin, T. (1989) Empirical Bayes Methods, Chapman & Hall, London.

    MATH  Google Scholar 

  • Masry, E. (1994) Probability density estimation from dependent observations using wavelet orthonormal bases. Statist. Probab. Lett., 21, 181–194.

    Article  MathSciNet  MATH  Google Scholar 

  • Meyer, I. (1993) Wavelets. Algorithms and Applications, SIAM, Philadelphia.

    MATH  Google Scholar 

  • Nogami, Y. (1988) Convergence rates for empirical Bayes estimation in the uniform U(0,θ) distribution. Ann. Stat, 16, 1335–1341.

    Article  MathSciNet  MATH  Google Scholar 

  • Penskaya, M. (1995) On mean square consistent estimation of a ratio. Scandinavian Journal of Statistics, 22, 129–137.

    MathSciNet  MATH  Google Scholar 

  • Pensky, M. (1996) Empirical Bayes estimation of a scale parameter. Mathematical Methods of Statistics, 5, 316–331.

    MathSciNet  MATH  Google Scholar 

  • Pensky, M. (1997a) Empirical Bayes estimation of a location parameter. Statist. Decisions, 15, 1–16.

    MathSciNet  MATH  Google Scholar 

  • Pensky, M. (1997b) A general approach to nonparametric empirical Bayes estimation. Statistics, 29, 61–80.

    Article  MathSciNet  MATH  Google Scholar 

  • Pensky, M. (1997c) Wavelet estimation in empirical Bayes model. Proceedings of the Section on Bayesian Statistical Science, American Statistical Association, 78-82.

    Google Scholar 

  • Pensky, M. (1998a) Empirical Bayes estimation based on wavelets. Sankhyā, A60, 214–231.

    MathSciNet  Google Scholar 

  • Pensky, M. (1998b) Wavelet empirical Bayes estimation of a location or a scale parameter. Journal of Statistical Planning and Inference, submitted.

    Google Scholar 

  • Robbins, H. (1983) Jerzy Neyman memorial lecture. Some thoughts on empirical Bayes estimation. Ann. Statist., 11, 713–723.

    Article  MathSciNet  MATH  Google Scholar 

  • Singh, R.S. (1979) Empirical Bayes estimation in Lebesgue-exponential families with rates near the best possible rate. Ann.Statist., 7, 890–902.

    Article  MathSciNet  MATH  Google Scholar 

  • Singh, R.S., and Wei, L. (1992) Empirical Bayes with rates and best rates of convergence in u(x)c(θ)exp (-x/θ) family estimation case. Ann. Inst. Statist. Math., 44, 435–449.

    MathSciNet  MATH  Google Scholar 

  • Tiwari, R.C., and Zalkikar, J.N. (1990) Empirical Bayes estimation of a scale parameter in a Pareto distribution. Comput. Statist. & Data Anal, 10, 261–270.

    Article  MathSciNet  MATH  Google Scholar 

  • Walter, G.G. (1981) Orthogonal series estimators of the prior distribution. Sankhya, A43, 228–245.

    Google Scholar 

  • Walter, G.G. (1994) Wavelets and Other Orthogonal Systems with Applications, CRC Press, Boca Raton.

    MATH  Google Scholar 

  • Zayed, A.L, and Walter, G.G. (1996) Characterization of analytic functions in terms of their wavelet coefficients. Complex Variables, 29, 265–276.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Authors
  1. Marianna Pensky
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Statistics and Decision Sciences, Duke University, Box 90251, Durham, 27708-0251, NC, England

    Peter Müller  & Brani Vidakovic  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer Science+Business Media New York

About this chapter

Cite this chapter

Pensky, M. (1999). Nonparametric Empirical Bayes Estimation via Wavelets. In: Müller, P., Vidakovic, B. (eds) Bayesian Inference in Wavelet-Based Models. Lecture Notes in Statistics, vol 141. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0567-8_20

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-0567-8_20

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-98885-6

  • Online ISBN: 978-1-4612-0567-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation