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 in many cases violated. In this paper we propose to model the dependence between ε and X by means of a copula model, i.e. (ε,X) ∼ Cθ(Fε(·), FX(·)), where Cθ is a copula function depending on an unknown parameter θ, and Fε and FX 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. We estimate 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.