Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

The Choice of a Primary Resolution and Basis Functions in Wavelet Series for Random or Irregular Design Points Using Bayesian Methods

  • Published : 2008.05.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, the choice of a primary resolution and wavelet basis functions are introduced under random or irregular design points of which the sample size is free of a power of two. Most wavelet methods have used the number of the points as the primary resolution. However, it turns out that a proper primary resolution is much affected by the shape of an unknown function. The proposed methods are illustrated by some simulations.

Keywords

References

  1. Abramovich, F., Bailey, T. C. and Sapatinas, T. (2000). Wavelet analysis and its statistical applications, The Statistician, 49, 1-29
  2. Antoniadis, A., Bigot, J. and Sapatinas, T. (2001). Wavelet estimators in nonparametric regression: A comparative simulation study, Journal of Statistical Software, 6
  3. Antoniadis, A., Gregoire, G. and Vial, P. (1997). Random design wavelet curve smoothing, Statistics & Probability Letters, 35, 225-232 https://doi.org/10.1016/S0167-7152(97)00017-5
  4. Antoniadis, A. and Sapatinas, T. (2001). Wavelet shrinkage for natural exponential families with quadratic variance functions, Biometrika, 88, 805-820 https://doi.org/10.1093/biomet/88.3.805
  5. Cai, T. and Brown, L. D. (1998). Wavelet shrinkage for nonequispaced samples, The Annals of Statistics, 26, 1783-1799 https://doi.org/10.1214/aos/1024691357
  6. Daubechies, I. (1992). Ten Lectures on Wavelets. (CBMS-NSF regional conference series in applied mathematics), SIAM: Society for industrial and applied mathmatics, Philadelphia
  7. Hall, P. and Patil, P. (1995). Formulae for mean integrated squared error of nonlinear wavelet-based density estimators, The Annals of statistics, 23, 905-928 https://doi.org/10.1214/aos/1176324628
  8. HAardle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998). Wavelets, Approximation and Statistical Applications, (Lecture notes in statistics), 129, Springer, New York
  9. Kovac, A. and Silverman, B. W. (2000). Extending the scope of wavelet regression methods by coefficient-dependent thresholding, Journal of the American Statistical Association, 95, 172-183 https://doi.org/10.2307/2669536
  10. Mallat, S. G. (1989). A theory for multiresolution signal decomposition: The wavelet representation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 11, 674-693 https://doi.org/10.1109/34.192463
  11. Maxim, V. (2002). Denoising signals observed on a random design, Paper presented to the fifth AFA-SMAI conference on curves and surfaces
  12. Park, C. G., Oh, H. S. and Lee, H. (2008). Bayesian selection of primary resolution and wavelet basis functions for wavelet regression, Computational Statistics, 23, 291-302 https://doi.org/10.1007/s00180-007-0055-y
  13. Pensky, M. and Vidakovic, B. (2001). On non-equally spaced wavelet regression, Annals of the Institute of Statistical Mathematics, 53, 681-690 https://doi.org/10.1023/A:1014640632666
  14. Vidakovic, B. (1999). Linear versus nonlinear rules for mixture normal priors, Annals of Institute of Statistical Mathematics, 51, 111-124 https://doi.org/10.1023/A:1003835319163