Bernoulli
- Bernoulli
- Volume 10, Number 2 (2004), 187-220.
On minimax density estimation on \mathbb{R}}
Anatoli Juditsky and Sophie Lambert-Lacroix
Full-text: Open access
Abstract
The problem of density estimation on \mathbb{R}} on the basis of an independent sample X1,..., XN with common density f is discussed. The behaviour of the minimax Lp risk, 1≤p≤∞, is studied when f belongs to a Hölder class of regularity s on the real line. The lower bound for the minimax risk is given. We show that the linear estimator is not efficient in this setting and construct a wavelet adaptive estimator which attains (up to a logarithmic factor in N) the lower bounds involved. We show that the minimax risk depends on the parameter p when p<2+ 1/s.
Article information
Source
Bernoulli, Volume 10, Number 2 (2004), 187-220.
Dates
First available in Project Euclid: 19 April 2004
Permanent link to this document
https://projecteuclid.org/euclid.bj/1082380217
Digital Object Identifier
doi:10.3150/bj/1082380217
Mathematical Reviews number (MathSciNet)
MR2046772
Zentralblatt MATH identifier
1076.62037
Keywords
adaptive estimation minimax estimation nonparametric density estimation
Citation
Juditsky, Anatoli; Lambert-Lacroix, Sophie. On minimax density estimation on \mathbb{R}}. Bernoulli 10 (2004), no. 2, 187--220. doi:10.3150/bj/1082380217. https://projecteuclid.org/euclid.bj/1082380217
References
- [1] Anscombe, F. (1948) The transformation of Poisson, binomial and negative-binomial data. Biometrika, 35, 246-254.
- [2] Birgé, L. (1983) Approximation dans les espaces métriques et théorie de l´estimation. Z. Wahrsheinlichkeitstheorie Verw. Geb., 65, 181-237.Digital Object Identifier: doi:10.1007/BF00532480
- [3] Birgé, L. and Massart, P. (2000) An adaptive compression algorithm in Besov spaces. Constr. Approx., 16(1), 1-36.
- [4] Bretagnolle, J. and Huber, C. (1979) Estimation des densités: risque minimax. Z. Wahrscheinlichkeitstheorie Verw. Geb., 47, 119-137.Digital Object Identifier: doi:10.1007/BF00535278
- [5] Butucea, C. (2000) The adaptive rate of convergence in a problem of pointwise density estimation. Statist. Probab. Lett., 47(1), 85-90. Abstract can also be found in the ISI/STMA publication
- [6] Ciesielski, Z. and Kamont, A. (1999) On the choice of the spline density estimators. Math. Pannon., 10(1), 17-28.
- [7] Daubechies, I. (1992) Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math. 61. Philadelphia: Society for Industrial and Applied Mathematics.Mathematical Reviews (MathSciNet): MR1162107
- [8] Delyon, B and Juditsky, A. (1996) On minimax wavelet estimators. Appl. Comput. Harmon. Anal., 3(3), 215-228.
- [9] Delyon, B. and Juditsky, A. (1997) On the computation of wavelet coefficients. J. Approx. Theory, 88(1), 47-79.Mathematical Reviews (MathSciNet): MR1426459
Zentralblatt MATH: 0862.42023
Digital Object Identifier: doi:10.1006/jath.1996.3008 - [10] Devroye, L. (1987) A Course in Density Estimation, Progr. Probab. Statist. 14. Boston: Birkhäuser Verlag.
- [11] Devroye, L. and Györfi, L. (1985) Nonparametric Density Estimation. The L1 View. New York: Wiley.
- [12] Donoho, D. (1994) Smooth wavelet decompositions with blocky coefficient kernels. In L. Schumaker and G. Webb (eds), Recent Advances in Wavelet Analysis. Boston: Academic Press.
- [13] Donoho, D. and Johnstone, I. (1995) Adapting to unknown smoothness via wavelet shrinkage. J. Amer. Statist. Assoc., 90, 1200-1224. Abstract can also be found in the ISI/STMA publicationURL: Link to item
Mathematical Reviews (MathSciNet): MR1379464
Zentralblatt MATH: 0869.62024
Digital Object Identifier: doi:10.2307/2291512
JSTOR: links.jstor.org - [14] Donoho, D.L., Johnstone, I.M., Kerkyacharian, G. and Picard, D. (1995) Wavelet shrinkage: Asymptopia? J. Roy. Statist. Soc. Ser. B, 57, 301-369.
- [15] Donoho, D., Johnstone, I., Kerkyacharian, G. and Picard, D. (1996) Density estimation by wavelet thresholding. Ann. Statist., 24(2), 508-539. Abstract can also be found in the ISI/STMA publicationURL: Link to item
Mathematical Reviews (MathSciNet): MR1394974
Zentralblatt MATH: 0860.62032
Digital Object Identifier: doi:10.1214/aos/1032894451
Project Euclid: euclid.aos/1032894451 - [16] Goldenshluger, A. and Nemirovski, A. (1997) On spatially adaptive estimation of nonparametric regression. Math. Methods Statist., 6(2), 135-170. Abstract can also be found in the ISI/STMA publication
- [17] Golubev, G. (1992) Nonparametric estimation of smooth probability densities in L2. Problems Inform. Transmission, 28(1), 44-54.Mathematical Reviews (MathSciNet): MR1163140
- [18] Ibragimov, I. and Khas´minskij, R. (1980) On the estimation of the distribution density. Zap. Nauchn. Sem. Leningr. Otdel. Mat. Inst. Steklov., 98, 61-85.
- [19] Ibragimov, I. and Khas´minskij, R. (1981) Statistical Estimation. Asymptotic Theory. Berlin: Springer- Verlag.
- [20] Juditsky, A. (1997) Wavelet estimators: Adapting to unknown smoothness. Math. Methods Statist., 6(1), 1-25. Abstract can also be found in the ISI/STMA publication
- [21] Juditsky, A. and Nazin, A. (2001) On minimax approach to non-parametric adaptive control. Internat. J. Adapt. Control Signal Process., 15(2), 153-168. Abstract can also be found in the ISI/STMA publication
- [22] Korostelev, A. and Tsybakov, A. (1993) Minimax Theory of Image Reconstruction, Lecture Notes in Statist. 82. Berlin: Springer-Verlag.Mathematical Reviews (MathSciNet): MR1226450
- [23] Lepskij, O. (1992) Asymptotically minimax adaptive estimation. II: Schemes without optimal adaption: Adaptive estimators. Theory Probab. Appl., 37(3), 433-448.Mathematical Reviews (MathSciNet): MR1214353
- [24] Nemirovskii, A. (1986) Nonparametric estimation of smooth regression functions. J. Comput. System Sci., 23(6), 1-11.Mathematical Reviews (MathSciNet): MR844292
- [25] Nussbaum, M. (1996) Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist., 24, 2399-2430. Abstract can also be found in the ISI/STMA publicationURL: Link to item
Mathematical Reviews (MathSciNet): MR1425959
Zentralblatt MATH: 0867.62035
Digital Object Identifier: doi:10.1214/aos/1032181160
Project Euclid: euclid.aos/1032181160 - [26] Silverman, B. (1986) Density Estimation for Statistics and Data Analysis. London: Chapman & Hall.
- [27] Triebel, H. (1992) Theory of Function Spaces II, Monogr. Math. 84. Basel: Birkhäuser Verlag.
- You have access to this content.
- You have partial access to this content.
- You do not have access to this content.
More like this
- Adaptive estimation in the nonparametric random coefficients binary choice model by needlet thresholding
Gautier, Eric and Le Pennec, Erwan, Electronic Journal of Statistics, 2018 - Adaptive sup-norm estimation of the Wigner function in noisy quantum homodyne tomography
Lounici, Karim, Meziani, Katia, and Peyré, Gabriel, The Annals of Statistics, 2018 - On adaptive posterior concentration rates
Hoffmann, Marc, Rousseau, Judith, and Schmidt-Hieber, Johannes, The Annals of Statistics, 2015
- Adaptive estimation in the nonparametric random coefficients binary choice model by needlet thresholding
Gautier, Eric and Le Pennec, Erwan, Electronic Journal of Statistics, 2018 - Adaptive sup-norm estimation of the Wigner function in noisy quantum homodyne tomography
Lounici, Karim, Meziani, Katia, and Peyré, Gabriel, The Annals of Statistics, 2018 - On adaptive posterior concentration rates
Hoffmann, Marc, Rousseau, Judith, and Schmidt-Hieber, Johannes, The Annals of Statistics, 2015 - Global uniform risk bounds for wavelet deconvolution estimators
Lounici, Karim and Nickl, Richard, The Annals of Statistics, 2011 - Trade-offs between global and local risks in nonparametric function estimation
Cai, T. Tony, Low, Mark G., and Zhao, Linda H., Bernoulli, 2007 - Wavelet Optimal Estimations for Density Functions under Severely Ill-Posed Noises
Li, Rui and Liu, Youming, Abstract and Applied Analysis, 2013 - Exact asymptotics for estimating the marginal density of discretely observed diffusion processes
Butucea, Cristina and Neumann, Michael H., Bernoulli, 2005 - Anisotropic de-noising in functional deconvolution model with dimension-free convergence rates
Benhaddou, Rida, Pensky, Marianna, and Picard, Dominique, Electronic Journal of Statistics, 2013 - Nonparametric Laguerre estimation in the multiplicative censoring model
Belomestny, Denis, Comte, Fabienne, and Genon-Catalot, Valentine, Electronic Journal of Statistics, 2016 - Adaptive demixing in Poisson mixture models
Hengartner, Nicolas W., The Annals of Statistics, 1997