Flexible modeling based on copulas in nonparametric median regression

Consider the model Y=m(X)+@e, where m(@?)=med(Y|@?) is unknown but smooth. It is often assumed that @e and X are independent. However, in practice this assumption is violated in many cases. In this paper we propose modeling the dependence between @e and X by means of a copula model, i.e. (@e,X)~C"@q(F"@e(@?),F"X(@?)), where C"@q is a copula function depending on an unknown parameter @q, and F"@e and F"X are the marginals of @e 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 @q 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.

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