Regression in random design and Bayesian warped wavelets estimators
Thanh Mai Pham Ngoc
Source: Electron. J. Statist. Volume 3
(2009), 1084-1112.
Abstract
In this paper we deal with the regression problem in a random design setting. We investigate asymptotic optimality under minimax point of view of various Bayesian rules based on warped wavelets. We show that they nearly attain optimal minimax rates of convergence over the Besov smoothness class considered. Warped wavelets have been introduced recently, they offer very good computable and easy-to-implement properties while being well adapted to the statistical problem at hand. We particularly put emphasis on Bayesian rules leaning on small and large variance Gaussian priors and discuss their simulation performances, comparing them with a hard thresholding procedure.
First Page:
Show
Hide
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.ejs/1258380624
Digital Object Identifier: doi:10.1214/09-EJS466
Mathematical Reviews number (MathSciNet): MR2566182
References
[1] Abramovich, F., Amato, U. and C. Angelini. On optimality of Bayesian wavelet estimators., Scand. J. Statist., 31(2):217–234, 2004.
Mathematical Reviews (MathSciNet): MR2066250
Digital Object Identifier: doi:10.1111/j.1467-9469.2004.02-087.x
[2] Abramovich F., Sapatinas, T. and Silverman, B. W. Wavelet thresholding via a Bayesian approach., J. R. Stat. Soc. Ser. B Stat. Methodol., 60(4):725–749, 1998.
Mathematical Reviews (MathSciNet): MR1649547
Zentralblatt MATH: 0910.62031
Digital Object Identifier: doi:10.1111/1467-9868.00151
[3] Amato, U., Antoniadis, A. and Pensky, M. Wavelet kernel penalized estimation for non-equispaced design regression., Stat. Comput., 16(1):37–55, 2006.
Mathematical Reviews (MathSciNet): MR2224188
Digital Object Identifier: doi:10.1007/s11222-006-5283-4
[4] Antoniadis, A., Grégoire, G. and Vial, P. Random design wavelet curve smoothing. 35:225–232, 1997.
Mathematical Reviews (MathSciNet): MR1484959
[5] Autin, F., Picard, D. and Rivoirard, V. Large variance Gaussian priors in Bayesian nonparametric estimation: a maxiset approach., Math. Methods of Statist., 15(4):349–373, 2006.
Mathematical Reviews (MathSciNet): MR2301657
[6] Baraud, Y. Model selection for regression on a random design., ESAIM Probab. Statist., 6:127–146, 2002.
Mathematical Reviews (MathSciNet): MR1918295
Digital Object Identifier: doi:10.1051/ps:2002007
[7] Brutti, P. Warped wavelets and vertical thresholding. Preprint. arxiv, :0801.3319v1.
[8] Cai, T. T. and Brown, L. D. Wavelet shrinkage for nonequispaced samples., Ann. Statist., 26 :1783–1799, 1998.
Mathematical Reviews (MathSciNet): MR1673278
Zentralblatt MATH: 0929.62047
Digital Object Identifier: doi:10.1214/aos/1024691357
Project Euclid: euclid.aos/1024691357
[9] Chipman, H. A., Kolaczyk, E. D. and McCulloch, R. E. Adaptive Bayesian Wavelet Shrinkage., J. Amer. Statist. Assoc., 92 :1413–1421, 1997.
[10] Clyde, M. and George, E. I. Flexible Empirical Bayes Estimation for Wavelets., J. Roy. Statist. Soc., Ser. B., 62(4):681–698, 2000.
Mathematical Reviews (MathSciNet): MR1796285
Zentralblatt MATH: 0957.62006
Digital Object Identifier: doi:10.1111/1467-9868.00257
[11] Clyde, M., Parmigiani, G. and Vidakovic, B. Multiple shrinkage and subset selection in wavelets., Biometrika, 85(2):391–401, 1998.
Mathematical Reviews (MathSciNet): MR1649120
Zentralblatt MATH: 0938.62021
Digital Object Identifier: doi:10.1093/biomet/85.2.391
[12] Clyde, M. A. and George, E. I. Empirical Bayes estimation in wavelet nonparametric regression. In, Bayesian inference in wavelet-based models, volume 141 of Lecture Notes in Statist., pages 309–322. Springer, New York, 1999.
Mathematical Reviews (MathSciNet): MR1699849
Zentralblatt MATH: 0936.62008
Digital Object Identifier: doi:10.1007/978-1-4612-0567-8_19
[13] Donoho, D. L. and Johnstone, I. M. Ideal spatial adaptation by wavelet shrinkage., Biometrika, 81(3):425–455, 1994.
Mathematical Reviews (MathSciNet): MR1311089
Zentralblatt MATH: 0815.62019
Digital Object Identifier: doi:10.1093/biomet/81.3.425
[14] Donoho, D. L. and Johnstone, I. M. Adapting to unknown smoothness via wavelet shrinkage., J. Amer. Statist. Assoc., 90(432) :1200–1224, 1995.
Mathematical Reviews (MathSciNet): MR1379464
Zentralblatt MATH: 0869.62024
Digital Object Identifier: doi:10.1080/01621459.1995.10476626
[15] Donoho, D. L., Johnstone, I. M., Kerkyacharian, G. and Picard, D. Wavelet shrinkage: asymptopia?, J. Roy. Statist. Soc. Ser. B, 57(2):301–369, 1995. With discussion and a reply by the authors.
Mathematical Reviews (MathSciNet): MR1323344
[16] Gannaz, I., Estimation par ondelettes dans les modèles partiellement linéaires. Ph.D. Thesis, Université Joseph Fourier, 2007.
[17] Hall, P. and Turlach, B. A. Interpolation methods for nonlinear wavelet regression with irregularly spaced design., Ann. Statist., 25 :1912–1925, 1997.
Mathematical Reviews (MathSciNet): MR1474074
Zentralblatt MATH: 0881.62044
Digital Object Identifier: doi:10.1214/aos/1069362378
Project Euclid: euclid.aos/1069362378
[18] Johnstone, I. M. and Silverman, B. W. Needles and straw in haystacks: empirical Bayes estimates of possibly sparse sequences., Ann. Statist., 32(4) :1594–1649, 2004.
Mathematical Reviews (MathSciNet): MR2089135
Zentralblatt MATH: 1047.62008
Digital Object Identifier: doi:10.1214/009053604000000030
Project Euclid: euclid.aos/1091626180
[19] Johnstone, I. M. and Silverman, B. W. Empirical Bayes selection of wavelet thresholds., Ann. Statist., 33(4) :1700–1752, 2005.
Mathematical Reviews (MathSciNet): MR2166560
Zentralblatt MATH: 1078.62005
Digital Object Identifier: doi:10.1214/009053605000000345
Project Euclid: euclid.aos/1123250227
[20] Kerkyacharian, G. and Picard, D. Regression in random design and warped wavelets., Bernoulli, 10(6) :1053–1105, 2004.
Mathematical Reviews (MathSciNet): MR2108043
Digital Object Identifier: doi:10.3150/bj/1106314850
Project Euclid: euclid.bj/1106314850
[21] Kerkyacharian, G. and Picard, D. Thresholding algorithms, maxisets and well concentrated basis., Test, 9(2):283–344, 2004.
Mathematical Reviews (MathSciNet): MR1821645
Zentralblatt MATH: 1107.62323
Digital Object Identifier: doi:10.1007/BF02595738
[22] Kovac, A. and Silverman, B. W. Extending the scope of wavelet regression methods by coefficient-dependent thresholding., J. Amer. Statist. Assoc., 95:172–183, 2000.
[23] Massart, P., Concentration inequalities and model selection, volume 1896 of Lecture Notes in Mathematics. Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour, July 6–23, 2003.
Mathematical Reviews (MathSciNet): MR2319879
Zentralblatt MATH: 1170.60006
[24] Pensky, M. Frequentist optimality of Bayesian wavelet shrinkage rules for Gaussian and non-Gaussian noise., Ann. Statist., 34(2):769–807, 2006.
Mathematical Reviews (MathSciNet): MR2283392
Zentralblatt MATH: 1095.62049
Digital Object Identifier: doi:10.1214/009053606000000128
Project Euclid: euclid.aos/1151418240
[25] Rivoirard, V. Bayesian modeling of sparse sequences and maxisets for Bayes rules., Math. Methods Statist., 14(3):346–376, 2005.
Mathematical Reviews (MathSciNet): MR2195330
[26] Willer, T., Estimation non paramétrique et problèmes inverses. Ph.D. Thesis, Université Paris VII, 2006.
Electronic Journal of Statistics
