title: Isotonic Distributional Regression
Creators: name: Henzi, Alexander
Publisher: name: Universität Bern
Year: 2022
Subjects: subject: 510 Mathematics

description: Distributional regression estimates the probability distribution of a response variable conditional on covariates. The estimated conditional distribution comprehensively summarizes the available information on the response variable, and allows to derive all statistical quantities of interest, such as the conditional mean, threshold exceedance probabilities, or quantiles.

This thesis develops isotonic distributional regression, a method for estimating conditional distributions under the assumption of a monotone relationship between covariates and a response variable. The response variable is univariate and real-valued, and the covariates lie in a partially ordered set. The monotone relationship is formulated in terms of stochastic order constraints, that is, the response variable increases in a stochastic sense as the covariates increase in the partial order. This assumption alone yields a shape-constrained non-parametric estimator, which does not involve any tuning parameters.

The estimation of distributions under stochastic order restrictions has already been studied for various stochastic orders, but so far only with totally ordered covariates. Apart from considering more general partially ordered covariates, the first main contribution of this thesis lies in a shift of focus from estimation to prediction. Distributional regression is the backbone of probabilistic forecasting, which aims at quantifying the uncertainty about a future quantity of interest comprehensively in the form of probability distributions. When analyzed with respect to predominant criteria for probabilistic forecast quality, isotonic distributional regression is shown to have desirable properties. In addition, this thesis develops an efficient algorithm for the computation of isotonic distributional regression, and proposes an estimator under a weaker, previously not thoroughly studied stochastic order constraint.

A main application of isotonic distributional regression is the uncertainty quantification for point forecasts. Such point forecasts sometimes stem from external sources, like physical models or expert surveys, but often they are generated with statistical models. The second contribution of this thesis is the extension of isotonic distributional regression to allow covariates that are point predictions from a regression model, which may be trained on the same data to which isotonic distributional regression is to be applied. This combination yields a so-called distributional index model. Asymptotic consistency is proved under suitable assumptions, and real data applications demonstrate the usefulness of the method.

Isotonic distributional regression provides a benchmark in forecasting problems, as it allows to quantify the merits of a specific, tailored model for the application at hand over a generic method which only relies on monotonicity. In such comparisons it is vital to assess the significance of forecast superiority or of forecast misspecification. The third contribution of this thesis is the development of new, safe methods for forecast evaluation, which require no or minimal assumptions on the data generating processes.