Abstract
We consider the nonparametric estimation problem of conditional regression quantiles with high-dimensional covariates. For the additive quantile regression model, we propose a new procedure such that the estimated marginal effects of additive conditional quantile curves do not cross. The method is based on a combination of the marginal integration technique and non-increasing rearrangements, which were recently introduced in the context of estimating a monotone regression function. Asymptotic normality of the estimates is established with a one-dimensional rate of convergence and the finite sample properties are studied by means of a simulation study and a data example.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Belsley D.A., Kuh E., Welsch R.E. (1980) Regression diagnostics. Identifying influential data and sources of collinearity. Wiley, New York
Bennett C., Sharpley R. (1988) Interpolation of operators. Academic Press, NY
Chen, R., Härdle, W., Linton, O. B., Severence-Lossin, E. (1996). Nonparametric estimation of additive separable regression models. Statistical theory and computational aspects of smoothing (Semmering, 1994) (pp. 247–265).
Chernozhukov, V., Fernandez-Val, I., Galichon, A. (2007). Quantile and probability curves without crossing. arXiv:0704.3649. http://trefoil.math.ucdavis.edu/0704.3649.
Collomb G., Härdle W. (1986) Strong uniform convergence rates in robust nonparametric time series analysis and prediction: Kernel regression estimation from dependent observations. Stochastic Processes and their Application 23(1): 77–89
De Gooijer J.G., Zerom D. (2003) On additive conditional quantiles with high-dimensional covariates. Journal of the American Statistical Association 98(461): 135–146
Dette H., Volgushev S. (2008) Non-crossing nonparametric estimates of quantile curves. Journal of the Royal Statistical Society: Series B 70(3): 609–627
Dette H., Neumeyer N., Pilz K.F. (2006) A simple nonparametric estimator of a strictly monotone regression function. Bernoulli 12: 469–490
Doksum K., Koo J.Y. (2000) On spline estimators and prediction intervals in nonparametric regression. Computational Statistics & Data Analysis 35: 67–82
Fan J., Gijbels I. (1996) Local polynomial modelling and its applications. Chapman and Hall, London
Hall P., Wolff R.C.L., Yao Q. (1999) Methods for estimating a conditional distribution function. Journal of the American Statistical Association 94(445): 154–163
Harrison D., Rubinfeld D.L. (1978) Hedonic prices and the demand for clean air. Journal of Environmental Economics and Management 5: 81–102
He X. (1997) Quantile curves without crossing. The American Statistician 51(2): 186–192
Hengartner N.W., Sperlich S. (2005) Rate optimal estimation with the integration method in the presence of many covariates. Journal of Multivariate Analysis 95(2): 246–272
Horowitz J., Lee S. (2005) Nonparametric estimation of an additive quantile regression model. Journal of the American Statistical Association 100(472): 1238–1249
Koenker R. (2005) Quantile regression. Cambridge University Press, London
Koenker R., Bassett G. (1978) Regression quantiles. Econometrica 46(1): 33–50
Linton O., Nielsen J.P. (1995) A Kernel method of estimating structured nonparametric regression based on marginal integration. Biometrika 82(1): 93–100
Masry E., Fan J. (1997) Local polynomial estimation of regression functions for mixing processes. Scandinavian Journal of Statistics 24(2): 165–179
Neumeyer N., Sperlich S. (2006) Comparison of separable components in different sample. Scandinavian Journal of Statistics 33: 477–501
Yu K., Jones M.C. (1997) A comparison of local constant and local linear regression quantile estimators. Computational Statistics & Data Analysis 25(2): 159–166
Yu K., Jones M.C. (1998) Local linear quantile regression. Journal of the American Statistical Association 93(441): 228–237
Yu K., Lu Z., Stander J. (2003) Quantile regression: applications and current research areas. The Statistician 52(3): 331–350
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Dette, H., Scheder, R. Estimation of additive quantile regression. Ann Inst Stat Math 63, 245–265 (2011). https://doi.org/10.1007/s10463-009-0225-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-009-0225-5