Hostname: page-component-5c6d5d7d68-thh2z Total loading time: 0 Render date: 2024-08-14T19:09:31.887Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniform Confidence Bands for Local Polynomial Quantile Estimators∗

Published online by Cambridge University Press:  25 July 2014

Camille Sabbah
Camille Sabbah*
Affiliation:
Laboratoire EQUIPPE, Université Lille Nord de France, Domaine Universitaire du Pont de Bois, BP 60149, 59653 Villeneuve-d’Ascq Cedex, France. camille.sabbah@univ-lille3.fr
Get access

Abstract

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].

Keywords

Uniform confidence bandsconditional quantile estimation
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 265 - 276
Copyright
© EDP Sciences, SMAI, 2014

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

Bahadur, R.R., A note on quantiles in large samples. Ann. Math. Stat. 37 (1966) 577–580. Google Scholar
Bickel, P.J. and Rosenblatt, M., On some global measures of the deviation of density function estimates. Ann. Statist. 1 (1973) 1071–1095. Google Scholar
Claeskens, G. and Van Keilegom, I., Bootstrap confidence bands for regression curves and their derivatives. Ann. Statist. 31 (2003) 1852–1884. Google Scholar
Einmahl, U. and Mason, D.M., Uniform in bandwidth consistency of kernel–type function estimators. Ann. Statist. 3 (2005) 1380–1403. Google Scholar
Eubanck, R.L. and Speckman, P.L., Confidence bands in nonparametric regression. J. Amer. Stat. Associat. 88 (1993) 1287–1301. Google Scholar
Fan, J. and Gijbels, I., Local Polynomial Modeling And Its Applications. Monogr. Stat. Appl. Prob. Chapman and Hall 66 (1996). Google Scholar
Guerre, E. and Sabbah, C., Uniform bias study and Bahadur representation for local polynomial estimators of the conditional quantile function. Econom. Theory. 28 (2012) 87–129. Google Scholar
Härdle, W., Asymptotic maximal deviation of M–smoothers. J. Mult. Anal. 29 (1989) 163–179. Google Scholar
Härdle, W., Ritov, Y. and Song, S., Partial linear quantile regression and bootstrap confidence bands. J. Mult. Ana. 107 (2012) 244–262. Google Scholar
Härdle, W. and Song, S., The Stochastic fluctuation of the quantile regression curve. Econom. Theory 26 (2010) 1180–1200. Google Scholar
Huber, P.J., Robust estimation of a location parameter. Ann. Math. Stat. 37 (1964) 73–101. Google Scholar
P.J. Huber, Robust Statistics. Wiley Series in Probab. Math. Statist. John Wiley and Sons, Inc., New York (1981).
Knafl, G., Sacks, J. and Ylvisaker, D., Confidence bands for regression functions. J. Amer. Stat. Associat. 80 (1985) 683–691. Google Scholar
R. Koenker, Quantile Regression. New York, Cambridge University Press (2005).
Koenker, R. and Basset, G., Regression quantiles. Econometrica 46 (1978) 33–50. Google Scholar
Kong, E., Linton, O. and Xia, Y., Uniform Bahadur representation for local polynomial estimates of M–regression and its application to the additive model. Econom. Theory. 26 (2010) 159–166. Google Scholar
Leung, D.H.-Y., Cross–validation in nonparametric regression with outliers. Ann. Statist. 33 (2005) 2291–2310. Google Scholar
Li, Q. and Racine, J.S., Nonparametric estimation of conditional CDF and quantile function with mixed categorical and continuous data. J. Busin. Econ. Statist. 26 (2008) 423–434. Google Scholar
R. Maronna, D. Martin and V. Yohai, Robust statistics, theory and methods. Wiley (2006).
Powell, J.L., Censored regression quantiles. J. Econom. 32 (1986) 143–155. Google Scholar
Rosenblatt, M., Remarks on a multivariate transformation. Ann. Math. Stat. 23 (1952) 470-472. Google Scholar
Stone, C.J., Optimal global rates of convergence for nonparametric regression. Ann. Statist. 10 (1982) 1040–1053. Google Scholar
Sun, J. and Loader, C.R., Simultaneous confidence bands for linear regression and smoothing. Ann. Statist. 22 (1994) 1328–1345. Google Scholar
Tusnàdy, G., A remark on the approximation of the sample distribution function in the multidimensional case. Period. Math. Hungar. 8 (1977) 53-55. Google Scholar
Wang, J. and Yang, L., Polynomial spline confidence bands for regression curves. Statistica Sinica. 19 (2009) 325–342. Google Scholar
Yu, K. and Jones, M.C., Local Linear Quantile Regression. J. Amer. Stat. Associat. 93 (1998) 228–237. Google Scholar