Geodesic
One Bootstrap suffices to generate sharp uniform bounds in functional estimation
Kybernetika, Tome 47 (2011) no. 6, pp. 855-865 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

We consider, in the framework of multidimensional observations, nonparametric functional estimators, which include, as special cases, the Akaike–Parzen–Rosenblatt kernel density estimators ([1, 18, 20]), and the Nadaraya–Watson kernel regression estimators ([16, 22]). We evaluate the sup-norm, over a given set ${\bf I}$, of the difference between the estimator and a non-random functional centering factor (which reduces to the estimator mean for kernel density estimation). We show that, under suitable general conditions, this random quantity is consistently estimated by the sup-norm over ${\bf I}$ of the difference between the original estimator and a bootstrapped version of this estimator. This provides a simple and flexible way to evaluate the estimator accuracy, through a single bootstrap. The present work generalizes former results of Deheuvels and Derzko [4], given in the setup of density estimation in $\mathbb{R}$.
We consider, in the framework of multidimensional observations, nonparametric functional estimators, which include, as special cases, the Akaike–Parzen–Rosenblatt kernel density estimators ([1, 18, 20]), and the Nadaraya–Watson kernel regression estimators ([16, 22]). We evaluate the sup-norm, over a given set ${\bf I}$, of the difference between the estimator and a non-random functional centering factor (which reduces to the estimator mean for kernel density estimation). We show that, under suitable general conditions, this random quantity is consistently estimated by the sup-norm over ${\bf I}$ of the difference between the original estimator and a bootstrapped version of this estimator. This provides a simple and flexible way to evaluate the estimator accuracy, through a single bootstrap. The present work generalizes former results of Deheuvels and Derzko [4], given in the setup of density estimation in $\mathbb{R}$.
Classification : 62G05, 62G08, 62G09, 62G15, 62G20, 62G30
Keywords: nonparametric functional estimation; density estimation; regression estimation; bootstrap; resampling methods; confidence regions; empirical processes 
@article{KYB_2011_47_6_a3,
     author = {Deheuvels, Paul},
     title = {One {Bootstrap} suffices to generate sharp uniform bounds in functional estimation},
     journal = {Kybernetika},
     pages = {855--865},
     year = {2011},
     volume = {47},
     number = {6},
     mrnumber = {2907846},
     zbl = {06047590},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2011_47_6_a3/}
}
TY  - JOUR
AU  - Deheuvels, Paul
TI  - One Bootstrap suffices to generate sharp uniform bounds in functional estimation
JO  - Kybernetika
PY  - 2011
SP  - 855
EP  - 865
VL  - 47
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/KYB_2011_47_6_a3/
LA  - en
ID  - KYB_2011_47_6_a3
ER  - 
%0 Journal Article
%A Deheuvels, Paul
%T One Bootstrap suffices to generate sharp uniform bounds in functional estimation
%J Kybernetika
%D 2011
%P 855-865
%V 47
%N 6
%U http://geodesic.mathdoc.fr/item/KYB_2011_47_6_a3/
%G en
%F KYB_2011_47_6_a3
Deheuvels, Paul. One Bootstrap suffices to generate sharp uniform bounds in functional estimation. Kybernetika, Tome 47 (2011) no. 6, pp. 855-865. http://geodesic.mathdoc.fr/item/KYB_2011_47_6_a3/

[1] Akaike, H.: An approximation of the density function. Ann. Inst. Statist. Math. 6 (1954), 127–132. | DOI | MR

[2] Bickel, P. J., Rosenblatt, M.: On some global measures of the deviations of density functions estimates. Ann. Statist. 1 (1973), 1071–1095. | DOI | MR

[3] Claeskens, G., Keilegom, I. van: Bootstrap confidence bands for regression curves and their derivatives. Ann. Statist. 31 (2003), 1852–1884. | DOI | MR

[4] Deheuvels, P., Derzko, G.: Asymptotic certainty bands for kernel density estimators based upon a bootstrap resampling scheme. In: Statistical Models and Methods for Biomedical and Technical systems, Stat. Ind. Technol., Birkhäuser, Boston MA 2008, pp. 171–186. | MR

[5] Deheuvels, P., Mason, D. M.: General asymptotic confidence bands based on kernel-type function estimators. Statist. Infer. Stoch. Processes 7 (2004), 225–277. | DOI | MR | Zbl

[6] Dudley, R. M.: Uniform Central Limit Theorems. Cambridge University Press, Cambridge 1999. | MR | Zbl

[7] Efron, B.: Bootstrap methods: Another look at the jackknife. Ann. Statist. 7 (1979), 1–26. | DOI | MR | Zbl

[8] Einmahl, U., Mason, D. M.: Uniform in bandwidth consistency of kernel-type function estimators. Ann. Statist. 33 (2005), 1380–1403. | DOI | MR | Zbl

[9] Giné, E., Koltchinskii, V., Sakhanenko, L.: Kernel density estimators: convergence in distribution for weighted sup-norms. Probab. Theory Related Fields 130 (2004), 167–198. | DOI | MR | Zbl

[10] Giné, E., Mason, D.  M.: On local $U$-statistic processes and the estimation of densities of functions of several sample variables. Ann. Statist. 35 (2007), 1105–1145. | DOI | MR | Zbl

[11] Hall, P.: The Bootstrap and Edgeworth Expansion. Springer, New York 1992. | MR | Zbl

[12] Härdle, W., Marron, J. S.: Bootstrap simultaneous error bars for nonparametric regression. Ann. Statist. 19 (1991), 778–796. | DOI | MR

[13] Härdle, W., Bowman, A. W.: Bootstrapping in nonparametric regression: Local adaptative smoothing and confidence bands. J. Amer. Statist. Assoc. 83 (1988), 102–110. | MR

[14] Li, G., Datta, S.: A bootstrap approach to nonparametric regression for right censored data. Ann. Inst. Statist. Math. 53 (2001), 708–729. | DOI | MR | Zbl

[15] Mason, D.  M., Newton, M. A.: A rank statistics approach to the consistency of a general bootstrap. Ann. Statist. 20 (1992), 1611–1624. | DOI | MR | Zbl

[16] Nadaraya, E. A.: On estimating regression. Theor. Probab. Appl. 9 (1964), 141–142. | DOI | Zbl

[17] Nolan, D., Pollard, D.: $U$-processes: Rates of convergence. Ann. Statist. 15 (1987), 780–799. | DOI | MR | Zbl

[18] Parzen, E.: On the estimation of probability density and mode. Ann. Math. Statist. 33 (1962), 1065–1076. | DOI | MR

[19] Pollard, D.: Convergence of Stochastic Processes. Springer, New York 1984. | MR | Zbl

[20] Rosenblatt, M.: Remarks on some nonparametric estimates of a density function. Ann. Math. Statist. 27 (1956), 832–837. | DOI | MR | Zbl

[21] Vaart, A. W. van der, Wellner, J. A.: Weak Convergence and Empirical Processes with Aplications to Statistics. Springer, New York 1996. | MR

[22] Watson, G. S.: Smooth regression analysis. Sankhyā A 26 (1964), 359–372. | MR | Zbl