Open Access
April 1997 On average derivative quantile regression
Probal Chaudhuri, Kjell Doksum, Alexander Samarov
Ann. Statist. 25(2): 715-744 (April 1997). DOI: 10.1214/aos/1031833670


For fixed $\alpha \epsilon (0, 1)$, the quantile regression function gives the $\alpha$th quantile $\theta_{\alpha}(\mathbf{x})$ in the conditional distribution of a response variable Y given the value $\mathbf{X} = \mathbf{x}$ of a vector of covariates. It can be used to measure the effect of covariates not only in the center of a population, but also in the upper and lower tails. A functional that summarizes key features of the quantile specific relationship between X and Y is the vector $\beta_{\alpha}$ of weighted expected values of the vector of partial derivatives of the quantile function $\theta_{\alpha}(\mathbf{x})$. In a nonparametric setting, $\beta_{\alpha}$ can be regarded as a vector of quantile specific nonparametric regression coefficients. In survival analysis models (e.g., Cox's proportional hazard model, proportional odds rate model, accelerated failure time model) and in monotone transformation models used in regression analysis, $\beta_{\alpha}$ gives the direction of the parameter vector in the parametric part of the model. $\beta_{\alpha}$ can also be used to estimate the direction of the parameter vector in semiparametric single index models popular in econometrics. We show that, under suitable regularity conditions, the estimate of $\beta_{\alpha}$ obtained by using the locally polynomial quantile estimate of Chaudhuri (1991a) is $n^{1/2}$-consistent and asymptotically normal with asymptotic variance equal to the variance of the influence function of the functional $\beta_{\alpha}$. We discuss how the estimate of $\beta_{\alpha}$ can be used for model diagnostics and in the construction of a link function estimate in general single index models.


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Probal Chaudhuri. Kjell Doksum. Alexander Samarov. "On average derivative quantile regression." Ann. Statist. 25 (2) 715 - 744, April 1997.


Published: April 1997
First available in Project Euclid: 12 September 2002

zbMATH: 0898.62082
MathSciNet: MR1439320
Digital Object Identifier: 10.1214/aos/1031833670

Primary: 62J02
Secondary: 62G99

Keywords: Average derivative estimate , Heteroscedasticity , index model , projection pursuit model , quantile specific regression coefficients , reduction of dimensionality , Survival analysis , transformation model

Rights: Copyright © 1997 Institute of Mathematical Statistics

Vol.25 • No. 2 • April 1997
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