Skip to main content


  • Emmanuel Guerre (a1) and Camille Sabbah (a2)

This paper investigates the bias and the weak Bahadur representation of a local polynomial estimator of the conditional quantile function and its derivatives. The bias and Bahadur remainder term are studied uniformly with respect to the quantile level, the covariates, and the smoothing parameter. The order of the local polynomial estimator can be higher than the differentiability order of the conditional quantile function. Applications of the results deal with global optimal consistency rates of the local polynomial quantile estimator, performance of random bandwidths, and estimation of the conditional quantile density function. The latter allows us to obtain a simple estimator of the conditional quantile function of the private values in a first-price sealed bids auction under the independent private values paradigm and risk neutrality.

Corresponding author
*Address corrspondence to Emmanuel Guerre, School of Economics and Finance, Queen Mary University of London, E1 4N5; London, Great Britain; e-mail:
Hide All
Chauduri, P. (1991) Nonparametric estimates of regression quantiles and their local Bahadur representation. Annals of Statistics 19, 760777.
Chernozhukov, V. & Hansen, C. (2005) An IV model of quantile treatments effects. Econometrica 73, 245261.
Chesher, A. (2003) Identification in nonseparable models. Econometrica 71, 14051441.
Chow, Y.S. & Teicher, H. (2003) Probability Theory. Independence, Interchangeability Martingales, 3rd ed.Springer-Verlag.
Echenique, F. & Komunjer, I. (2009) Testing models with multiple equilibria by quantile methods. Econometrica 77, 12811297.
Einmahl, U. & Mason, D.M. (2005) Uniform in bandwidth consistency of kernel-type function estimators. Annals of Statistics 33, 13801403.
Fan, J. (1992) Design-adaptive nonparametric regression. Journal of the American Statistical Association 87, 9981004.
Fan, J. & Gijbels, I. (1996) Local Polynomial Modeling and Its Applications. Chapman and Hall/CRC.
Fan, J., Heckman, N.E., & Wand, M.P. (1995) Local polynomial kernel regression for generalized linear model and quasi-likelihood functions. Journal of the American Statistical Association 90, 141151.
Firpo, S., Fortin, N., & Lemieux, T. (2009) Unconditional quantile regression. Econometrica 77, 953973.
Goldenshluger, A. & Lepski, O. (2008) Universal pointwise selection rule in multivariate function estimation. Bernoulli 14, 11501190.
Goldenshluger, A. & Lepski, O. (2009) Structural adaptation via -norm oracle inequalities. Probability Theory and Related Fields 143, 4171.
Guerre, E., Perrigne, I., & Vuong, Q. (2000) Optimal nonparametric estimation of first price auctions. Econometrica 68, 525574.
Guerre, E., Perrigne, I., & Vuong, Q. (2009) Nonparametric identification of risk aversion in first-price auctions under exclusion restrictions. Econometrica 77, 11931227.
Haile, P.A., Hong, H., & Shum, M. (2003) Nonparametric Tests for Common Values in First-Price Sealed-Bid Auctions. Cowles Foundation Discussion paper.
Hjort, N. & Pollard, D. (1993) Asymptotics for Minimisers of Convex Processes. Manuscript, Yale.
Holderlein, S. & Mammen, E. (2007) Identification of marginal effects in nonseparable models without monotonicity. Econometrica 75, 15131518.
Holderlein, S. & Mammen, E. (2009) Identification and estimation of local average derivatives in non-separable models without monotonicity. Econometrics Journal 12, 125.
Hong, S.Y. (2003) Bahadur representation and its applications for local polynomial estimates in nonparametric M-regression. Journal of Nonparametric Statistics 15, 237251.
Imbens, G.W. & Newey, W.K. (2009) Identification and estimation of triangular simultaneous equations models without additivity. Econometrica 77, 14811512.
Koenker, R. (2005) Quantile Regression. Cambridge University Press.
Kong, E., Linton, O., & Xia, Y. (2010) Uniform Bahadur representation for local polynomial estimates of M-regression and its application to the additive model. Econometric Theory 26, 15291564.
Lee, K.L. & Lee, E.R. (2008) Kernel methods for estimating derivatives of conditional quantiles. Journal of the Korean Statistical Society 37, 365373.
Li, Q. & Racine, J.S. (2008) Nonparametric estimation of conditional CDF and quantile functions with mixed categorical and continuous data. Journal of Business & Economic Statistics 26, 423434.
Loader, C. (1999) Local Regression and Likelihood. Springer-Verlag.
Marmer, V. & Shneyerov, A. (2012) Quantile-based nonparametric inference for first-price auctions. Journal of Econometrics, in press.
Massart, P. (2007) Concentration Inequalities and Model Selection. Lecture Notes in Mathematics 1896. Ecole d’Eté de Probabilités de Saint Flour XXXIII-2003, Picard, Jean (ed.). Springer-Verlag.
Parzen, E. (1979) Nonparametric statistical data modeling. Journal of the American Statistical Association 74, 105121.
Rothe, C. (2010) Nonparametric estimation of distributional policy effects. Journal of Econometrics 155, 5670.
Stone, C.J. (1982) Optimal global rates of convergence for nonparametric regression. Annals of Statistics 10, 10401053.
Truong, Y.K. (1989) Asymptotic properties of kernel estimators based on local medians. Annals of Statistics 17, 606617.
Tsybakov, A.B. (1986) Robust reconstruction of functions by the local-approximation method. Problemy Peredachi Informatsii 22, 6984.
van de Geer, S. (1999) Empirical Processes in M-Estimation. Cambridge University Press.
van der Vaart, A.W. (1998) Asymptotic Statistics. Cambridge University Press.
White, H. (1994) Estimation, Inference and Specification Analysis. Econometric Society Monographs. Cambridge University Press.
Xiang, X. (1995) Estimation of conditional quantile density function. Journal of Nonparametric Statistics 4, 309316.
Zeidler, E. (1985) Nonlinear Functional Analysis and Its Applications, Vol. 1: Fixed-Point Theorems. Springer-Verlag.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed