Nonparametric regression expectiles∗

Abstract

It is well known that a standard nonparametric regression analysis is to model the average behavior between the dependent variable Y and the explanatory variable x. But such an approach may not always be appropriate if one is interested in the extreme behavior of Y conditional on x. This paper considers the problem of estimating the expectile function of the conditional distribution of YY given x based on the observational data generated according to a nonparametric regression model. We proposed a kernel-type nonparametric regression estimator, called nonparametric regression expectile, using an asymmetric squared loss function. This estimator models not only the average behavior but also the extreme behavior of Y given x in the nonparametric regression setting. An iterative algorithm is presented to calculate the estimator. It is shown that the nonparametric regression expectile is consistent and asymptotically normally distributed. We also derive a lower bound for the asymptotic variance and the asympto...