Adaptive non-asymptotic confidence balls in density estimation
ESAIM: Probability and Statistics, Tome 16 (2012), pp. 61-85.

We build confidence balls for the common density s of a real valued sample X1,...,Xn. We use resampling methods to estimate the projection of s onto finite dimensional linear spaces and a model selection procedure to choose an optimal approximation space. The covering property is ensured for all n ≥ 2 and the balls are adaptive over a collection of linear spaces.

DOI : https://doi.org/10.1051/ps/2010012
Classification : 62G07,  62G09,  62G10,  62G15
Mots clés : confidence balls, density estimation, resampling methods
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author = {Lerasle, Matthieu},
title = {Adaptive non-asymptotic confidence balls in density estimation},
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Lerasle, Matthieu. Adaptive non-asymptotic confidence balls in density estimation. ESAIM: Probability and Statistics, Tome 16 (2012), pp. 61-85. doi : 10.1051/ps/2010012. http://archive.numdam.org/articles/10.1051/ps/2010012/

[1] S. Arlot, Model selection by resampling penalization. Electron. J. Statist. 3 (2009) 557-624. | MR 2519533

[2] S. Arlot and P. Massart, Data-driven calibration of penalties for least-squares regression. J. Mach. Learn. Res. 10 (2009) 245-279.

[3] S. Arlot, G. Blanchard and E. Roquain, Resampling-based confidence regions and multiple tests for a correlated random vector, in Learning theory. Lect. Notes Comput. Sci. 4539 (2007) 127-141. | MR 2397583 | Zbl 1203.62030

[4] Y. Baraud, Confidence balls in Gaussian regression. Ann. Statist. 32 (2004) 528-551. | MR 2060168 | Zbl 1093.62051

[5] R. Beran, REACT scatterplot smoothers : superefficiency through basis economy. J. Amer. Statist. Assoc. 95 (2000) 155-171. | MR 1803148 | Zbl 1013.62073

[6] R. Beran and L. Dümbgen, Modulation of estimators and confidence sets. Ann. Statist. 26 (1998) 1826-1856. | MR 1673280 | Zbl 1073.62538

[7] L. Birgé and P. Massart, From model selection to adaptive estimation, in Festschrift for Lucien Le Cam. Springer, New York (1997) 55-87. | MR 1462939 | Zbl 0920.62042

[8] L. Birgé and P. Massart, Minimal penalties for Gaussian model selection. Probab. Theory Relat. Fields 138 (2007) 33-73. | MR 2288064 | Zbl 1112.62082

[9] T. Cai and M.G. Low, Adaptive confidence balls. Ann. Statist. 34 (2006) 202-228. | MR 2275240 | Zbl 1091.62037

[10] B. Efron, Bootstrap methods : another look at the jackknife. Ann. Statist. 7 (1979) 1-26. | MR 515681 | Zbl 0406.62024

[11] M. Fromont and B. Laurent, Adaptive goodness-of-fit tests in a density model. Ann. Statist. 34 (2006) 680-720. | MR 2281881 | Zbl 1096.62040

[12] C.R. Genovese and L. Wasserman, Confidence sets for nonparametric wavelet regression. Ann. Statist. 33 (2005) 698-729. | MR 2163157 | Zbl 1068.62057

[13] C. Genovese and L. Wasserman, Adaptive confidence bands. Ann. Statist. 36 (2008) 875-905. | MR 2396818 | Zbl 1139.62311

[14] M. Hoffmann and O. Lepski, Random rates in anisotropic regression. Ann. Statist. 30 (2002) 325-396. With discussions and a rejoinder by the authors. | MR 1902892 | Zbl 1012.62042

[15] C. Houdré and P. Reynaud-Bouret, Exponential inequalities, with constants, for U-statistics of order two, in Stochastic inequalities and applications. Progr. Probab. 56 (2003) 55-69. | MR 2073426 | Zbl 1036.60015

[16] Y.I. Ingster, Asymptotically minimax hypothesis testing for nonparametric alternatives. I. Math. Methods Stat. 2 (1993) 85-114. | MR 1257978 | Zbl 0798.62057

[17] Y.I. Ingster, Asymptotically minimax hypothesis testing for nonparametric alternatives. II. Math. Methods Stat. 2 (1993) 171-189. | MR 1257983 | Zbl 0798.62058

[18] Y.I. Ingster, Asymptotically minimax hypothesis testing for nonparametric alternatives. III. Math. Methods Stat. 2 (1993) 249-268. | MR 1259685 | Zbl 0798.62059

[19] A. Juditsky and S. Lambert-Lacroix, Nonparametric confidence set estimation. Math. Methods Stat. 12 (2003) 410-428. | MR 2054156

[20] A. Juditsky and O. Lepski, Evaluation of the accuracy of nonparametric estimators. Math. Methods Stat. 10 (2001) 422-445. Meeting on Mathematical Statistics, Marseille (2000). | MR 1887341 | Zbl 1005.62041

[21] B. Laurent, Estimation of integral functionnals of a density. Ann. Statist. 24 (1996) 659-681. | MR 1394981 | Zbl 0859.62038

[22] B. Laurent, Adaptive estimation of a quadratic functional of a density by model selection. ESAIM : PS 9 (2005) 1-18 (electronic). | Numdam | MR 2148958 | Zbl 1136.62333

[23] O.V. Lepski, How to improve the accuracy of estimation. Math. Methods Stat. 8 (1999) 441-486. | MR 1755896 | Zbl 1033.62032

[24] M. Lerasle, Optimal model selection in density estimation. Preprint (2009). | Numdam | MR 2976568 | Zbl 1244.62052

[25] K.C. Li, Honest confidence regions for nonparametric regression. Ann. Statist. 17 (1989) 1001-1008. | MR 1015135 | Zbl 0681.62047

[26] M.G. Low, On nonparametric confidence intervals. Ann. Statist. 25 (1997) 2547-2554. | MR 1604412 | Zbl 0894.62055

[27] P. Massart, Concentration inequalities and model selection. Springer, Berlin. Lect. Notes Math. 1896 (2007). Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour (2003). With a foreword by Jean Picard. | MR 2319879 | Zbl 1170.60006

[28] J. Robins and A. Van Der Vaart, Adaptive nonparametric confidence sets. Ann. Statist. 34 (2006) 229-253. | MR 2275241 | Zbl 1091.62039

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