Nonparametric Estimation of Global Functionals of Conditional Quantiles
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 \(\sqrt n\) 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.
Key words and phrasesAverage derivative estimate transformation model projection pursuit model index model heteroscedasticity reduction of dimensionality quantile specific regression coefficients
AMS 1991 subject classificationsPrimary 62J02 secondary 62G99.
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