Adaptive non-asymptotic confidence balls in density estimation

Matthieu Lerasle

ESAIM: Probability and Statistics (2012)

  • Volume: 16, page 61-85
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Lerasle, Matthieu. "Adaptive non-asymptotic confidence balls in density estimation." ESAIM: Probability and Statistics 16 (2012): 61-85. <http://eudml.org/doc/273612>.

@article{Lerasle2012,
abstract = {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.},
author = {Lerasle, Matthieu},
journal = {ESAIM: Probability and Statistics},
keywords = {confidence balls; density estimation; resampling methods},
language = {eng},
pages = {61-85},
publisher = {EDP-Sciences},
title = {Adaptive non-asymptotic confidence balls in density estimation},
url = {http://eudml.org/doc/273612},
volume = {16},
year = {2012},
}

TY - JOUR
AU - Lerasle, Matthieu
TI - Adaptive non-asymptotic confidence balls in density estimation
JO - ESAIM: Probability and Statistics
PY - 2012
PB - EDP-Sciences
VL - 16
SP - 61
EP - 85
AB - 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.
LA - eng
KW - confidence balls; density estimation; resampling methods
UR - http://eudml.org/doc/273612
ER -

References

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  1. [1] S. Arlot, Model selection by resampling penalization. Electron. J. Statist.3 (2009) 557–624. Zbl1326.62097MR2519533
  2. [2] S. Arlot and P. Massart, Data-driven calibration of penalties for least-squares regression. J. Mach. Learn. Res.10 (2009) 245–279. 
  3. [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. Zbl1203.62030MR2397583
  4. [4] Y. Baraud, Confidence balls in Gaussian regression. Ann. Statist.32 (2004) 528–551. Zbl1093.62051MR2060168
  5. [5] R. Beran, REACT scatterplot smoothers : superefficiency through basis economy. J. Amer. Statist. Assoc.95 (2000) 155–171. Zbl1013.62073MR1803148
  6. [6] R. Beran and L. Dümbgen, Modulation of estimators and confidence sets. Ann. Statist.26 (1998) 1826–1856. Zbl1073.62538MR1673280
  7. [7] L. Birgé and P. Massart, From model selection to adaptive estimation, in Festschrift for Lucien Le Cam. Springer, New York (1997) 55–87. Zbl0920.62042MR1462939
  8. [8] L. Birgé and P. Massart, Minimal penalties for Gaussian model selection. Probab. Theory Relat. Fields138 (2007) 33–73. Zbl1112.62082MR2288064
  9. [9] T. Cai and M.G. Low, Adaptive confidence balls. Ann. Statist.34 (2006) 202–228. Zbl1091.62037MR2275240
  10. [10] B. Efron, Bootstrap methods : another look at the jackknife. Ann. Statist.7 (1979) 1–26. Zbl0406.62024MR515681
  11. [11] M. Fromont and B. Laurent, Adaptive goodness-of-fit tests in a density model. Ann. Statist.34 (2006) 680–720. Zbl1096.62040MR2281881
  12. [12] C.R. Genovese and L. Wasserman, Confidence sets for nonparametric wavelet regression. Ann. Statist.33 (2005) 698–729. Zbl1068.62057MR2163157
  13. [13] C. Genovese and L. Wasserman, Adaptive confidence bands. Ann. Statist.36 (2008) 875–905. Zbl1139.62311MR2396818
  14. [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. Zbl1012.62042MR1902892
  15. [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. Zbl1036.60015MR2073426
  16. [16] Y.I. Ingster, Asymptotically minimax hypothesis testing for nonparametric alternatives. I. Math. Methods Stat.2 (1993) 85–114. Zbl0798.62057MR1257978
  17. [17] Y.I. Ingster, Asymptotically minimax hypothesis testing for nonparametric alternatives. II. Math. Methods Stat.2 (1993) 171–189. Zbl0798.62058MR1257983
  18. [18] Y.I. Ingster, Asymptotically minimax hypothesis testing for nonparametric alternatives. III. Math. Methods Stat.2 (1993) 249–268. Zbl0798.62059MR1259685
  19. [19] A. Juditsky and S. Lambert-Lacroix, Nonparametric confidence set estimation. Math. Methods Stat.12 (2003) 410–428. MR2054156
  20. [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). Zbl1005.62041MR1887341
  21. [21] B. Laurent, Estimation of integral functionnals of a density. Ann. Statist.24 (1996) 659–681. Zbl0859.62038MR1394981
  22. [22] B. Laurent, Adaptive estimation of a quadratic functional of a density by model selection. ESAIM : PS 9 (2005) 1–18 (electronic). Zbl1136.62333MR2148958
  23. [23] O.V. Lepski, How to improve the accuracy of estimation. Math. Methods Stat.8 (1999) 441–486. Zbl1033.62032MR1755896
  24. [24] M. Lerasle, Optimal model selection in density estimation. Preprint (2009). Zbl1244.62052MR2976568
  25. [25] K.C. Li, Honest confidence regions for nonparametric regression. Ann. Statist.17 (1989) 1001–1008. Zbl0681.62047MR1015135
  26. [26] M.G. Low, On nonparametric confidence intervals. Ann. Statist.25 (1997) 2547–2554. Zbl0894.62055MR1604412
  27. [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. Zbl1170.60006MR2319879
  28. [28] J. Robins and A. van der Vaart, Adaptive nonparametric confidence sets. Ann. Statist.34 (2006) 229–253. Zbl1091.62039MR2275241

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