Flexible modeling based on copulas in nonparametric median regression

Consider the model Y=m(X)+εY=m(X)+ε, where m(⋅)=med(Y|⋅)m(⋅)=med(Y|⋅) is unknown but smooth. It is often assumed that εε and XX are independent. However, in practice this assumption is violated in many cases. In this paper we propose modeling the dependence between εε and XX by means of a copula model, i.e. (ε,X)∼Cθ(Fε(⋅),FX(⋅))(ε,X)∼Cθ(Fε(⋅),FX(⋅)), where CθCθ is a copula function depending on an unknown parameter θθ, and FεFε and FXFX are the marginals of εε and XX. Since many parametric copula families contain the independent copula as a special case, the so-obtained regression model is more flexible than the ‘classical’ regression model.We estimate the parameter θθ via a pseudo-likelihood method and prove the asymptotic normality of the estimator, based on delicate empirical process theory. We also study the estimation of the conditional distribution of YY given XX. The procedure is illustrated by means of a simulation study, and the method is applied to data on food expenditures in households.