Distributional vs. Quantile Regression

Given a scalar random variable Y and a random vector X defined on the same probability space, the conditional distribution of Y given X can be represented by either the conditional distribution function or the conditional quantile function. To these equivalent representations correspond two alternative approaches to estimation. One approach, distributional regression (DR), is based on direct estimation of the conditional distribution function; the other approach, quantile regression (QR), is instead based on direct estimation of the conditional quantile function. Indirect estimates of the conditional quantile function and the conditional distribution function may then be obtained by inverting the direct estimates obtained from either approach. Despite the growing attention to the DR approach, and the vast literature on the QR approach, the link between the two approaches has not been explored in detail. The aim of this paper is to fill-in this gap by providing a better understanding of the relative performance of the two approaches, both asymptotically and in finite samples, under the linear location model and certain types of heteroskedastic location-scale models