Some results on generalized regression quantiles
In this paper, we obtain an asymptotic representation of generalized regression quantiles under various conditions. In particular, we consider a special family of loss functions. It is shown that by using these kind of loss functions, substantial improvement in efficiency can be achieved when the error distribution has one light and one heavy tail. We also discuss the scale equivariant properties of the estimators. It is found that a data-based method of choosing loss functions will automatically make the estimators scale equivariant. Our results are obtained under mild conditions and cover many results in the literature.