🤖 AI Summary
This paper addresses the lack of a rigorous theoretical foundation for consistency of scoring functions under variable transformations, specifically examining conditions for consistency and identifiability when predictions and observations undergo one-sided or bijective transformations.
Method: We establish formal necessary and sufficient conditions for (strict) consistency and identifiability under general transformations, integrating scoring function theory, Bregman divergence analysis, and techniques from elicitation and identification function characterization for expectation-like functionals. We introduce novel identifiable functionals—including the *g-transformed expectation* and *g-transformed quantile*—and analyze their elicitation properties.
Contribution/Results: Our framework provides the first unified theoretical justification for transformed scoring functions in empirical modeling. It enables principled construction of interpretable and verifiable functionals, with broad applicability to probabilistic forecasting and robust regression. The results bridge theoretical statistics and practical model evaluation, ensuring that transformation-based scoring remains both statistically sound and operationally meaningful.
📝 Abstract
Scoring functions constructed by transforming the realization and prediction variables of (strictly) consistent scoring functions have been widely studied empirically, yet their theoretical foundations remain unexplored. To address this gap, we establish formal characterizations of (strict) consistency for these transformed scoring functions and their elicitable functionals. Our analysis focuses on two interrelated cases: (a) transformations applied exclusively to the realization variable, and (b) bijective transformations applied jointly to both realization and prediction variables. We formulate analogous characterizations for (strict) identification functions. The resulting theoretical framework is broadly applicable to statistical and machine learning methodologies. When applied to Bregman and expectile scoring functions, our framework shows how it enables two critical advances: (a) rigorous interpretation of prior empirical findings from models trained with transformed scoring functions, and (b) systematic construction of novel identifiable and elicitable functionals, specifically the g-transformed expectation and g-transformed expectile. By unifying theoretical insights with practical applications, this work advances principled methodologies for designing scoring functions in complex predictive tasks.