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.
Mots-clés : confidence balls, density estimation, resampling methods
@article{PS_2012__16__61_0, author = {Lerasle, Matthieu}, title = {Adaptive non-asymptotic confidence balls in density estimation}, journal = {ESAIM: Probability and Statistics}, pages = {61--85}, publisher = {EDP-Sciences}, volume = {16}, year = {2012}, doi = {10.1051/ps/2010012}, mrnumber = {2946120}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ps/2010012/} }
TY - JOUR AU - Lerasle, Matthieu TI - Adaptive non-asymptotic confidence balls in density estimation JO - ESAIM: Probability and Statistics PY - 2012 SP - 61 EP - 85 VL - 16 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2010012/ DO - 10.1051/ps/2010012 LA - en ID - PS_2012__16__61_0 ER -
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] Model selection by resampling penalization. Electron. J. Statist. 3 (2009) 557-624. | MR
,[2] Data-driven calibration of penalties for least-squares regression. J. Mach. Learn. Res. 10 (2009) 245-279.
and ,[3] Resampling-based confidence regions and multiple tests for a correlated random vector, in Learning theory. Lect. Notes Comput. Sci. 4539 (2007) 127-141. | MR | Zbl
, and ,[4] Confidence balls in Gaussian regression. Ann. Statist. 32 (2004) 528-551. | MR | Zbl
,[5] REACT scatterplot smoothers : superefficiency through basis economy. J. Amer. Statist. Assoc. 95 (2000) 155-171. | MR | Zbl
,[6] Modulation of estimators and confidence sets. Ann. Statist. 26 (1998) 1826-1856. | MR | Zbl
and ,[7] From model selection to adaptive estimation, in Festschrift for Lucien Le Cam. Springer, New York (1997) 55-87. | MR | Zbl
and ,[8] Minimal penalties for Gaussian model selection. Probab. Theory Relat. Fields 138 (2007) 33-73. | MR | Zbl
and ,[9] Adaptive confidence balls. Ann. Statist. 34 (2006) 202-228. | MR | Zbl
and ,[10] Bootstrap methods : another look at the jackknife. Ann. Statist. 7 (1979) 1-26. | MR | Zbl
,[11] Adaptive goodness-of-fit tests in a density model. Ann. Statist. 34 (2006) 680-720. | MR | Zbl
and ,[12] Confidence sets for nonparametric wavelet regression. Ann. Statist. 33 (2005) 698-729. | MR | Zbl
and ,[13] Adaptive confidence bands. Ann. Statist. 36 (2008) 875-905. | MR | Zbl
and ,[14] Random rates in anisotropic regression. Ann. Statist. 30 (2002) 325-396. With discussions and a rejoinder by the authors. | MR | Zbl
and ,[15] Exponential inequalities, with constants, for U-statistics of order two, in Stochastic inequalities and applications. Progr. Probab. 56 (2003) 55-69. | MR | Zbl
and ,[16] Asymptotically minimax hypothesis testing for nonparametric alternatives. I. Math. Methods Stat. 2 (1993) 85-114. | MR | Zbl
,[17] Asymptotically minimax hypothesis testing for nonparametric alternatives. II. Math. Methods Stat. 2 (1993) 171-189. | MR | Zbl
,[18] Asymptotically minimax hypothesis testing for nonparametric alternatives. III. Math. Methods Stat. 2 (1993) 249-268. | MR | Zbl
,[19] Nonparametric confidence set estimation. Math. Methods Stat. 12 (2003) 410-428. | MR
and ,[20] Evaluation of the accuracy of nonparametric estimators. Math. Methods Stat. 10 (2001) 422-445. Meeting on Mathematical Statistics, Marseille (2000). | MR | Zbl
and ,[21] Estimation of integral functionnals of a density. Ann. Statist. 24 (1996) 659-681. | MR | Zbl
,[22] Adaptive estimation of a quadratic functional of a density by model selection. ESAIM : PS 9 (2005) 1-18 (electronic). | Numdam | MR | Zbl
,[23] How to improve the accuracy of estimation. Math. Methods Stat. 8 (1999) 441-486. | MR | Zbl
,[24] Optimal model selection in density estimation. Preprint (2009). | Numdam | MR | Zbl
,[25] Honest confidence regions for nonparametric regression. Ann. Statist. 17 (1989) 1001-1008. | MR | Zbl
,[26] On nonparametric confidence intervals. Ann. Statist. 25 (1997) 2547-2554. | MR | Zbl
,[27] 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 | Zbl
,[28] Adaptive nonparametric confidence sets. Ann. Statist. 34 (2006) 229-253. | MR | Zbl
and ,Cité par Sources :