Bootstrapping the shorth for regression
ESAIM: Probability and Statistics, Tome 10 (2006), pp. 216-235.

The paper is concerned with the asymptotic distributions of estimators for the length and the centre of the so-called $\eta$-shorth interval in a nonparametric regression framework. It is shown that the estimator of the length 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 location of 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.

DOI : https://doi.org/10.1051/ps:2006007
Classification : 62E20,  62G05,  62G08,  62G09
Mots clés : brownian motion with parabolic drift, bootstrap, location of maximum, shorth
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author = {Durot, C\'ecile and Thi\'ebot, Karelle},
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Durot, Cécile; Thiébot, Karelle. Bootstrapping the shorth for regression. ESAIM: Probability and Statistics, Tome 10 (2006), pp. 216-235. doi : 10.1051/ps:2006007. http://www.numdam.org/articles/10.1051/ps:2006007/

[1] 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). | MR 331595 | Zbl 0254.62001

[2] D. De Angelis, P. Hall and G.A. Young, Analytical and bootstrap approximations to estimator distributions in ${l}^{1}$ regression. J. Am. Stat. Assoc. 88 (1993) 1310-1316. | Zbl 0792.62029

[3] C. Durot and K. Thiébot, Detecting atypical data in air pollution studies by using shorth intervals for regression. ESAIM: PS 9 (2005) 230-240. | Numdam | Zbl 1137.62408

[4] M. Falk and R.-D. Reiss, Weak convergence of smoothed and nonsmoothed bootstrap quantiles estimates. Ann. Probab. 17 (1989) 362-371. | Zbl 0684.62036

[5] P. Groeneboom, Brownian motion with a parabolic drift and Airy functions Probab. Th. Rel. Fields 81 (1989) 79-109.

[6] R. Grübel, The length of the shorth. Ann. Statist. 16 (1988) 619-628. | Zbl 0664.62040

[7] P. Hall, Theoretical comparison of bootstrap confidence intervals. Ann. Statist. 16 (1988) 927-953. | Zbl 0663.62046

[8] P. Hall, T.J. Diciccio and J.P. Romano, On smoothing and the bootstrap. Ann. Statist. 17 (1989) 692-704. | Zbl 0672.62051

[9] P. Hall, J.W. Kay and D.M. Titterington, Asymptotically optimal difference-based estimation of variance in nonparametric regression. Biometrika 77 (1990) 521-528.

[10] E. Janaszewska and A.V. Nagaev, On the joint distribution of the shorth height and length. Math. Meth. Statist. 7 (1998) 210-229. | Zbl 1103.62350

[11] J. Kim and D. Pollard, Cube root asymptotics. Ann. Statist. 18 (1990) 191-219. | Zbl 0703.62063

[12] A. Narayanan and T.W. Sager, Table for the asymptotic distribution of univariate mode estimators. J. Stat. Comput. Simul. 33 (1989) 37-51. | Zbl 0726.62022

[13] A.I. Sakhanenko, Estimates in the invariance principle. Predel'nye Teoremy Teorii Veroyatnostej, Tr. Inst. Mat. 5 (1985) 27-44. | Zbl 0585.60044

[14] G.R. Shorack and J.A. Wellner, Empirical processes with applications to statistics. New York, Wiley (1986). | MR 838963

[15] 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. | Zbl 0591.62031

[16] K. Thiébot, Analyses statistiques et validation de données de pollution atmosphérique. Ph.D. thesis, Université Paris-Sud Orsay, France (2001).

[17] Y.G. Yatracos, On the estimation of the derivatives of a function with the derivatives of an estimate. J. Multivariate Anal. 28 (1989) 172-175. | Zbl 0665.62041

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