Nonparametric Estimation of Global Functionals of Conditional Quantiles

Abstract

For fixed α ∈ (0,1), the quantile regression function gives the αth quantile θ α (x) in the conditional distribution of a response variable Y given the value X = 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. When there are many covariates, the curse of dimensionality makes accurate estimation of the quantile regression function difficult. A functional that escapes this curse, at least asymptotically, and summarizes key features of the quantile specific relationship between X and Y is the vector β α of weighted expected values of the vector of partial derivatives of the quantile function θ α(x). In a nonparametric setting, β α can be regarded as the vector of quantile specific nonparametric regression coefficients while in semiparametric transformation and single index models, β α gives the direction of the parameter vector in the parametric part of the model. We show that, under suitable regularity conditions, the estimate of β α obtained by using the locally polynomial quantile estimate of Chaudhuri (1991a), is n−−√ consistent and asymptotically normal with asymptotic variance equal to the variance of the influence function of the functional β α. We discuss how the estimates of β α can be used for model diagnostics and in the construction of a link function estimates in general single index models.