Introduction of the asymptotic study of the estimation of the error distribution in right censored and selection biased regression models

Consider the regression model Y = m(X) + σ(X) Ɛ where m(X) = E[Y|X] and σ²(X)=Var[Y|X] are unknown smooth functions and the error Ɛ , with unknown distribution, is independent of the covariate X. The pair (X;Y) is subject to generalized bias selection and the response to right censoring. We construct a new estimator for the cumulative distribution function of the error Ɛ , where the estimators of m(.) and σ²(.) are obtained by extending the conditional estimation methods introduced in de Uña-Alvarez and Iglesias-Perez (2008). The asymptotic properties of the functions m(.) and σ(.) are obtained. A bootstrap technique is proposed to select the smoothing parameter involved in the procedure. This method is studied via extended simulations and applied to real unemployment data.