Distributional Regression

Abstract

This thesis aims to estimate conditional distribution functions subject to the likelihood ratio order constraint. We use the modified gradient projection method to ensure that in each iteration, the point stays feasible while improving the objective function. Regarding the objective function, we use the continuous ranked probability score (CRPS), a loss function used for forecast evaluation. Given its strict propriety, we use it for estimation procedures. Our numerical experiments indicate that the estimated conditional distribution functions perform reasonably well as the number of iteration increases. However, the algorithm’s long running time makes it impractical for use in practice. Furthermore, due to the reparametrization of the estimand, the objective function loses its convexity property while the feasible set is convex. This causes the algorithm to potentially return a local minimum, rather than a global minimum.