Power transformation toward a linear regression quantile

In this article we consider the linear quantile regression model with a power transformation on the dependent variable. Like the classical Box-Cox transformation approach, it extends the applicability of linear models without resorting to nonparametric smoothing, but transformations on the quantile models are more natural due to the equivariance property of the quantiles under monotone transformations. We propose an estimation procedure and establish its consistency and asymptotic normality under some regularity conditions. The objective function employed in the estimation can also be used to check inadequacy of a power-transformed linear quantile regression model and to obtain inference on the transformation parameter. The proposed approach is shown to be valuable through illustrative examples.