title: On Isotonic Conditional Laws and Sequential Forecast Evaluation
Creators: name: Arnold, Sebastian
Publisher: name: Universität Bern
Year: 2024
Subjects: subject: 360 Social problems &amp; social services
subject: 510 Mathematics

description: The conditional law of a response variable given some covariate represents the distribution of the variable of interest given all information about the explanatory random variable. In statistical terms, the conditional law describes the theoretically optimal probabilistic prediction for the distribution of the response having observed the covariate.
This thesis develops isotonic conditional laws by extending the classical notion of conditional laws under the additional requirement that there exists a monotone relationship between the response and the covariate. The monotone relationship is understood in the sense that large (small) realizations of the covariate make it more likely that the response will be large (small) as well, an assumption which is reasonable in many situations and is at the heart of isotonic regression. Isotonic conditional laws build on conditional expectations given σ-lattices and provide a population version for isotonic distributional regression. From a statistical perspective, isotonic conditional laws form theoretically optimal probabilistic predictions as calibrated minimizers of an expected score.
In the second part of the thesis, isotonic conditional laws are used to decompose the continuous ranked probability score, the most commonly used scoring rule to evaluate probabilistic forecasts for real-valued outcomes. A probabilistic prediction is a probability measure on the possible outcomes and incorporates the intrinsic uncertainty of the future event. Probabilistic forecasts should be evaluated by proper scoring rules, which assign a numerical score to a probabilistic forecast given the corresponding observation. In practice, different forecasters are compared according to the mean empirical scores assigned to their issued forecasts and the observed realizations. While proper scoring rules address the most important quality criteria for probabilistic forecasts, namely calibration and sharpness, simultaneously, a decomposition of the assessed score into more interpretable terms measuring miscalibration, discrimination ability and the overall uncertainty of the outcome, reveals a deeper insight into the performance of the assessed forecasts than the overall score on its own. The isotonicity-based decomposition of the continuous ranked probability score developed in this thesis allows for a rigorous theoretical analysis based on isotonic conditional laws and has various superior properties compared to other decompositions proposed in the literature.
The third contribution of this thesis is the development of new, sequential methods for forecast evaluation. Classically, two or more statistical models are compared with respect to their mean empirical scores over a fixed evaluation period. Similarly, probabilistic calibration of predictive distributions is traditionally assessed with respect to a prespecified number of observations. However, in the vast majority of cases, these tasks are faced in sequential situations, where models might adapt progressively and new data points are collected successively over time. This thesis provides methods to sequentially monitor probabilistic calibration and to sequentially monitor the best models in a given set of statistical models requiring no or only minimal assumptions on the data generating process and achieving anytime-valid guarantees.