Isotonic distributional regression

Isotonic distributional regression (IDR) is a powerful nonparametric technique for the estimation of conditional distributions under order restrictions. In a nutshell, IDR learns conditional distributions that are calibrated, and simultaneously optimal relative to comprehensive classes of relevant loss functions, subject to isotonicity constraints in terms of a partial order on the covariate space. Nonparametric isotonic quantile regression and probabilistic classifiers for binary events emerge as special cases. For prediction, we propose an interpolation method that generalizes extant specifications under the pool adjacent violators algorithm. We recommend the use of IDR as a generic benchmark technique in probabilistic forecast problems, as it does not involve any parameter tuning nor implementation choices, except for the selection of a partial order on the covariate space. The method can be combined with subsample aggregation, with the benefits of smoother regression functions and computational efficiency under large training sets. In a case study on raw and postprocessed quantitative precipitation forecasts from a leading numerical weather prediction system, IDR is competitive with state of the art techniques.