Open Access
April 2005 Extremal quantile regression
Victor Chernozhukov
Ann. Statist. 33(2): 806-839 (April 2005). DOI: 10.1214/009053604000001165


Quantile regression is an important tool for estimation of conditional quantiles of a response Y given a vector of covariates X. It can be used to measure the effect of covariates not only in the center of a distribution, but also in the upper and lower tails. This paper develops a theory of quantile regression in the tails. Specifically, it obtains the large sample properties of extremal (extreme order and intermediate order) quantile regression estimators for the linear quantile regression model with the tails restricted to the domain of minimum attraction and closed under tail equivalence across regressor values. This modeling setup combines restrictions of extreme value theory with leading homoscedastic and heteroscedastic linear specifications of regression analysis. In large samples, extreme order regression quantiles converge weakly to arg min functionals of stochastic integrals of Poisson processes that depend on regressors, while intermediate regression quantiles and their functionals converge to normal vectors with variance matrices dependent on the tail parameters and the regressor design.


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Victor Chernozhukov. "Extremal quantile regression." Ann. Statist. 33 (2) 806 - 839, April 2005.


Published: April 2005
First available in Project Euclid: 26 May 2005

zbMATH: 1068.62063
MathSciNet: MR2163160
Digital Object Identifier: 10.1214/009053604000001165

Primary: 62G30 , 62G32 , 62P20
Secondary: 62E30 , 62J05

Keywords: Conditional quantile estimation , Extreme value theory , regression

Rights: Copyright © 2005 Institute of Mathematical Statistics

Vol.33 • No. 2 • April 2005
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