Single-index quantile regression

Nonparametric quantile regression with multivariate covariates is a difficult estimation problem due to the “curse of dimensionality”. To reduce the dimensionality while still retaining the flexibility of a nonparametric model, we propose modeling the conditional quantile by a single-index function g0(xTγ0), where a univariate link function g0(⋅)g0(⋅) is applied to a linear combination of covariates xTγ0, often called the single-index. We introduce a practical algorithm where the unknown link function g0(⋅)g0(⋅) is estimated by local linear quantile regression and the parametric index is estimated through linear quantile regression. Large sample properties of estimators are studied, which facilitate further inference. Both the modeling and estimation approaches are demonstrated by simulation studies and real data applications.