Nonparametric Distributional Regression via Quantile Regression

This paper proposes a new approach to estimating the distribution of a response variable conditioned on factors. We model the conditional quantile function as a mixture (weighted sum) of basis quantile functions, with weights depending on these factors. The estimation problem is formulated as a convex optimization problem. The objective function is equivalent to the continuous ranked probability score (CRPS). This approach can be viewed as conducting quantile regressions for all confidence levels simultaneously while inherently avoiding quantile crossing. We use spline functions of factors as a primary example for the weight function. We prove an approximation property of the model. To address computational challenges, we propose a dimensionality reduction method using tensor decomposition and an alternating algorithm. Our approach offers flexibility, interpretability, tractability, and extendability. Numerical experiments demonstrate its effectiveness.