Flexible modeling based on copulas in nonparametric median regression

Consider the model Y = m ( X ) + ε , where m ( ⋅ ) = med ( Y | ⋅ ) is unknown but smooth. It is often assumed that ε and X are independent. However, in practice this assumption is violated in many cases. In this paper we propose modeling the dependence between ε and X by means of a copula model, i.e.  ( ε , X ) ∼ C θ ( F ε ( ⋅ ) , F X ( ⋅ ) ) , where C θ is a copula function depending on an unknown parameter θ , and F ε and F X are the marginals of ε and X . 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. imate the parameter θ 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 Y given X . The procedure is illustrated by means of a simulation study, and the method is applied to data on food expenditures in households.