High-dimensional quantile varying-coefficient models with dimension reduction

Although semiparametric models, in particular varying-coefficient models, alleviate the curse of dimensionality by avoiding estimation of fully nonparametric multivariate functions, there would typically still be a large number of functions to estimate. We propose a dimension reduction approach to estimating a large number of nonparametric univariate functions in varying-coefficient models, in which these functions are constrained to lie in a finite-dimensional subspace consisting of the linear span of a small number of smooth functions. The proposed methodology is put in the context of quantile regression, which provides more information on the response variable than the more conventional mean regression. Finally, we present some numerical illustrations to demonstrate the performances.