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Bootstrapping the shorth for regression

Published online by Cambridge University Press:  03 May 2006

Cécile Durot  and
Karelle Thiébot
Cécile Durot
Affiliation:
Université Paris-Sud, Bâtiment 425, 91405 Orsay Cedex, France; cecile.durot@math.u-psud.fr
Karelle Thiébot
Affiliation:
Université Paris-Sud, Bâtiment 425, 91405 Orsay Cedex, France; cecile.durot@math.u-psud.fr Air Pays de la Loire, 2 rue A. Kastler, BP 30723, 44307 Nantes Cedex 3, France. Supported by Air Pays De La Loire
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Abstract

The paper is concerned with the asymptotic distributions of estimators for the length and the centre of the so-called η-shorth interval in a nonparametric regression framework. It is shown that the estimator of thelength converges at the n 1/2-rate to a Gaussian law and that the estimator of the centre converges at the n 1/3-rate to the locationof the maximum of a Brownian motion with parabolic drift.Bootstrap procedures are proposed and shown to be consistent. They are compared with the plug-in method through simulations.

Keywords

Brownian motion with parabolic driftbootstraplocation of maximumshorth.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 10 , February 2006 , pp. 216 - 235
Copyright
© EDP Sciences, SMAI, 2006

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References

D.F. Andrews, P.J. Bickel, F.R. Hampel, P.J. Huber, W.H. Rogers and J.W. Tukey, Robust estimates of location. Survey and advances. Princeton Univ. Press, Princeton, N.J. (1972).
De Angelis, D., Hall, P. and Young, G.A., Analytical and bootstrap approximations to estimator distributions in l 1 regression. J. Am. Stat. Assoc. 88 (1993) 1310–1316.
Durot, C. and Thiébot, K., Detecting atypical data in air pollution studies by using shorth intervals for regression. ESAIM: PS 9 (2005) 230–240. CrossRef
Falk, M. and Reiss, R.-D., Weak convergence of smoothed and nonsmoothed bootstrap quantiles estimates. Ann. Probab. 17 (1989) 362–371. CrossRef
P. Groeneboom, Brownian motion with a parabolic drift and Airy functions Probab. Th. Rel. Fields 81 (1989) 79–109.
Grübel, R., The length of the shorth. Ann. Statist. 16 (1988) 619–628. CrossRef
P. Hall, Theoretical comparison of bootstrap confidence intervals. Ann. Statist. 16 (1988) 927–953.
Hall, P., DiCiccio, T.J. and Romano, J.P., On smoothing and the bootstrap. Ann. Statist. 17 (1989) 692–704. CrossRef
Hall, P., Kay, J.W. and Titterington, D.M., Asymptotically optimal difference-based estimation of variance in nonparametric regression. Biometrika 77 (1990) 521–528. CrossRef
Janaszewska, E. and Nagaev, A.V., On the joint distribution of the shorth height and length. Math. Meth. Statist. 7 (1998) 210–229.
Kim, J. and Pollard, D., Cube root asymptotics. Ann. Statist. 18 (1990) 191–219. CrossRef
Narayanan, A. and Sager, T.W., Table for the asymptotic distribution of univariate mode estimators. J. Stat. Comput. Simul. 33 (1989) 37–51. CrossRef
A.I. Sakhanenko, Estimates in the invariance principle. Predel'nye Teoremy Teorii Veroyatnostej, Tr. Inst. Mat. 5 (1985) 27–44.
G.R. Shorack and J.A. Wellner, Empirical processes with applications to statistics. New York, Wiley (1986).
C.J. Stone, Optimal uniform rate of convergence for nonparametric estimators of a density function and its derivatives. Recent Advances in Statistics, Academic Press, New York (1983) 293–406.
K. Thiébot, Analyses statistiques et validation de données de pollution atmosphérique. Ph.D. thesis, Université Paris-Sud Orsay, France (2001).
Yatracos, Y.G., On the estimation of the derivatives of a function with the derivatives of an estimate. J. Multivariate Anal. 28 (1989) 172–175. CrossRef