Hostname: page-component-7bb8b95d7b-lvwk9 Total loading time: 0 Render date: 2024-09-06T23:07:00.506Z Has data issue: false hasContentIssue false
  • English
  • Français

Adaptive non-asymptotic confidence balls in density estimation

Published online by Cambridge University Press:  02 July 2012

Matthieu Lerasle
Matthieu Lerasle*
Affiliation:
Institut de Mathématiques (UMR 5219), INSA de Toulouse, Université de Toulouse, France. [email protected]
Get access

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.

Keywords

Confidence ballsdensity estimationresampling methods
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 16 , 2012 , pp. 61 - 85
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Références

Arlot, S., Model selection by resampling penalization. Electron. J. Statist. 3 (2009) 557–624. Google Scholar
Arlot, S. and Massart, P., Data-driven calibration of penalties for least-squares regression. J. Mach. Learn. Res. 10 (2009) 245–279. Google Scholar
Arlot, S., Blanchard, G. and Roquain, E., Resampling-based confidence regions and multiple tests for a correlated random vector, in Learning theory. Lect. Notes Comput. Sci. 4539 (2007) 127–141. Google Scholar
Baraud, Y., Confidence balls in Gaussian regression. Ann. Statist. 32 (2004) 528–551. Google Scholar
Beran, R., REACT scatterplot smoothers : superefficiency through basis economy. J. Amer. Statist. Assoc. 95 (2000) 155–171. Google Scholar
Beran, R. and Dümbgen, L., Modulation of estimators and confidence sets. Ann. Statist. 26 (1998) 1826–1856. Google Scholar
L. Birgé and P. Massart, From model selection to adaptive estimation, in Festschrift for Lucien Le Cam. Springer, New York (1997) 55–87.
Birgé, L. and Massart, P., Minimal penalties for Gaussian model selection. Probab. Theory Relat. Fields 138 (2007) 33–73. Google Scholar
Cai, T. and Low, M.G., Adaptive confidence balls. Ann. Statist. 34 (2006) 202–228. Google Scholar
Efron, B., Bootstrap methods : another look at the jackknife. Ann. Statist. 7 (1979) 1–26. Google Scholar
Fromont, M. and Laurent, B., Adaptive goodness-of-fit tests in a density model. Ann. Statist. 34 (2006) 680–720. Google Scholar
Genovese, C.R. and Wasserman, L., Confidence sets for nonparametric wavelet regression. Ann. Statist. 33 (2005) 698–729. Google Scholar
Genovese, C. and Wasserman, L., Adaptive confidence bands. Ann. Statist. 36 (2008) 875–905. Google Scholar
Hoffmann, M. and Lepski, O., Random rates in anisotropic regression. Ann. Statist. 30 (2002) 325–396. With discussions and a rejoinder by the authors. Google Scholar
Houdré, C. and Reynaud-Bouret, P., Exponential inequalities, with constants, for U-statistics of order two, in Stochastic inequalities and applications. Progr. Probab. 56 (2003) 55–69. Google Scholar
Ingster, Y.I., Asymptotically minimax hypothesis testing for nonparametric alternatives. I. Math. Methods Stat. 2 (1993) 85–114. Google Scholar
Ingster, Y.I., Asymptotically minimax hypothesis testing for nonparametric alternatives. II. Math. Methods Stat. 2 (1993) 171–189. Google Scholar
Ingster, Y.I., Asymptotically minimax hypothesis testing for nonparametric alternatives. III. Math. Methods Stat. 2 (1993) 249–268. Google Scholar
Juditsky, A. and Lambert-Lacroix, S., Nonparametric confidence set estimation. Math. Methods Stat. 12 (2003) 410–428. Google Scholar
Juditsky, A. and Lepski, O., Evaluation of the accuracy of nonparametric estimators. Math. Methods Stat. 10 (2001) 422–445. Meeting on Mathematical Statistics, Marseille (2000). Google Scholar
Laurent, B., Estimation of integral functionnals of a density. Ann. Statist. 24 (1996) 659–681. Google Scholar
Laurent, B., Adaptive estimation of a quadratic functional of a density by model selection. ESAIM : PS 9 (2005) 1–18 (electronic). Google Scholar
Lepski, O.V., How to improve the accuracy of estimation. Math. Methods Stat. 8 (1999) 441–486. Google Scholar
M. Lerasle, Optimal model selection in density estimation. Preprint (2009).
Li, K.C., Honest confidence regions for nonparametric regression. Ann. Statist. 17 (1989) 1001–1008. Google Scholar
Low, M.G., On nonparametric confidence intervals. Ann. Statist. 25 (1997) 2547–2554. Google Scholar
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.
Robins, J. and van der Vaart, A., Adaptive nonparametric confidence sets. Ann. Statist. 34 (2006) 229–253. Google Scholar