Uniform Confidence Bands for Local Polynomial Quantile Estimators
ESAIM: Probability and Statistics, Tome 18 (2014), pp. 265-276.

This paper deals with uniform consistency and uniform confidence bands for the quantile function and its derivatives. We describe a kernel local polynomial estimator of quantile function and give uniform consistency. Furthermore, we derive its maximal deviation limit distribution using an approximation in the spirit of Bickel and Rosenblatt [P.J. Bickel and M. Rosenblatt, Ann. Statist. 1 (1973) 1071-1095].

DOI : 10.1051/ps/2013035
Classification : 62G08, 62G15
Mots clés : uniform confidence bands, conditional quantile estimation 
@article{PS_2014__18__265_0,
     author = {Sabbah, Camille},
     title = {Uniform {Confidence} {Bands} for {Local} {Polynomial} {Quantile} {Estimators}},
     journal = {ESAIM: Probability and Statistics},
     pages = {265--276},
     publisher = {EDP-Sciences},
     volume = {18},
     year = {2014},
     doi = {10.1051/ps/2013035},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps/2013035/}
}
TY  - JOUR
AU  - Sabbah, Camille
TI  - Uniform Confidence Bands for Local Polynomial Quantile Estimators
JO  - ESAIM: Probability and Statistics
PY  - 2014
SP  - 265
EP  - 276
VL  - 18
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps/2013035/
DO  - 10.1051/ps/2013035
LA  - en
ID  - PS_2014__18__265_0
ER  - 
%0 Journal Article
%A Sabbah, Camille
%T Uniform Confidence Bands for Local Polynomial Quantile Estimators
%J ESAIM: Probability and Statistics
%D 2014
%P 265-276
%V 18
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ps/2013035/
%R 10.1051/ps/2013035
%G en
%F PS_2014__18__265_0
Sabbah, Camille. Uniform Confidence Bands for Local Polynomial Quantile Estimators. ESAIM: Probability and Statistics, Tome 18 (2014), pp. 265-276. doi : 10.1051/ps/2013035. http://www.numdam.org/articles/10.1051/ps/2013035/

[1] R.R. Bahadur, A note on quantiles in large samples. Ann. Math. Stat. 37 (1966) 577-580. | MR | Zbl

[2] P.J. Bickel and M. Rosenblatt, On some global measures of the deviation of density function estimates. Ann. Statist. 1 (1973) 1071-1095. | MR | Zbl

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

[4] U. Einmahl and D.M. Mason, Uniform in bandwidth consistency of kernel-type function estimators. Ann. Statist. 3 (2005) 1380-1403. | MR | Zbl

[5] R.L. Eubanck and P.L. Speckman, Confidence bands in nonparametric regression. J. Amer. Stat. Associat. 88 (1993) 1287-1301. | MR | Zbl

[6] J. Fan and I. Gijbels, Local Polynomial Modeling And Its Applications. Monogr. Stat. Appl. Prob. Chapman and Hall 66 (1996). | MR | Zbl

[7] E. Guerre and C. Sabbah, Uniform bias study and Bahadur representation for local polynomial estimators of the conditional quantile function. Econom. Theory. 28 (2012) 87-129. | MR | Zbl

[8] W. Härdle, Asymptotic maximal deviation of M-smoothers. J. Mult. Anal. 29 (1989) 163-179. | Zbl

[9] W. Härdle, Y. Ritov and S. Song, Partial linear quantile regression and bootstrap confidence bands. J. Mult. Ana. 107 (2012) 244-262. | Zbl

[10] W. Härdle and S. Song, The Stochastic fluctuation of the quantile regression curve. Econom. Theory 26 (2010) 1180-1200. | Zbl

[11] P.J. Huber, Robust estimation of a location parameter. Ann. Math. Stat. 37 (1964) 73-101. | MR | Zbl

[12] P.J. Huber, Robust Statistics. Wiley Series in Probab. Math. Statist. John Wiley and Sons, Inc., New York (1981). | MR | Zbl

[13] G. Knafl, J. Sacks and D. Ylvisaker, Confidence bands for regression functions. J. Amer. Stat. Associat. 80 (1985) 683-691. | MR | Zbl

[14] R. Koenker, Quantile Regression. New York, Cambridge University Press (2005). | Zbl

[15] R. Koenker and G. Basset, Regression quantiles. Econometrica 46 (1978) 33-50. | MR | Zbl

[16] E. Kong, O. Linton and Y. Xia, Uniform Bahadur representation for local polynomial estimates of M-regression and its application to the additive model. Econom. Theory. 26 (2010) 159-166. | MR | Zbl

[17] D.H.-Y. Leung, Cross-validation in nonparametric regression with outliers. Ann. Statist. 33 (2005) 2291-2310. | MR | Zbl

[18] Q. Li and J.S. Racine, Nonparametric estimation of conditional CDF and quantile function with mixed categorical and continuous data. J. Busin. Econ. Statist. 26 (2008) 423-434. | MR

[19] R. Maronna, D. Martin and V. Yohai, Robust statistics, theory and methods. Wiley (2006). | MR | Zbl

[20] J.L. Powell, Censored regression quantiles. J. Econom. 32 (1986) 143-155. | MR | Zbl

[21] M. Rosenblatt, Remarks on a multivariate transformation. Ann. Math. Stat. 23 (1952) 470-472. | MR | Zbl

[22] C.J. Stone, Optimal global rates of convergence for nonparametric regression. Ann. Statist. 10 (1982) 1040-1053. | MR | Zbl

[23] J. Sun and C.R. Loader, Simultaneous confidence bands for linear regression and smoothing. Ann. Statist. 22 (1994) 1328-1345. | MR | Zbl

[24] G. Tusnàdy, A remark on the approximation of the sample distribution function in the multidimensional case. Period. Math. Hungar. 8 (1977) 53-55. | MR | Zbl

[25] J. Wang and L. Yang, Polynomial spline confidence bands for regression curves. Statistica Sinica. 19 (2009) 325-342. | MR | Zbl

[26] K. Yu and M.C. Jones, Local Linear Quantile Regression. J. Amer. Stat. Associat. 93 (1998) 228-237. | MR | Zbl

Cité par Sources :