ON AVERAGE DERIVATIVE QUANTILE ... - Project Euclid

The Annals of Statistics 1997, Vol. 25, No. 2, 715 ] 744

ON AVERAGE DERIVATIVE QUANTILE REGRESSION BY PROBAL CHAUDHURI,1 KJELL DOKSUM 2

AND

ALEXANDER SAMAROV 3

Indian Statistical Institute, University of California, Berkeley, and University of Massachusetts and MIT For fixed a g Ž0, 1., the quantile regression function gives the a th quantile ua Žx. in the conditional distribution of a response variable Y given the value X s 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 ba of weighted expected values of the vector of partial derivatives of the quantile function ua Žx.. In a nonparametric setting, ba can be regarded as a vector of quantile specific nonparametric regression coefficients. In survival analysis models Že.g., Cox’s proportional hazard model, proportional odds rate model, accelerated failure time model. and in monotone transformation models used in regression analysis, ba gives the direction of the parameter vector in the parametric part of the model. ba 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 ba obtained by using the locally polynomial quantile estimate of Chaudhuri Ž1991a. is n1r 2 -consistent and asymptotically normal with asymptotic variance equal to the variance of the influence function of the functional ba . We discuss how the estimate of ba can be used for model diagnostics and in the construction of a link function estimate in general single index models.

1. Introduction. The quantile regression function is defined as the a th quantile ua Žx. in the conditional distribution FY < X Ž y