Order-Sensitivity and Equivariance of Scoring Functions

TL;DR

This paper explores the properties of scoring functions used for forecast evaluation, focusing on order-sensitivity and equivariance, especially for vector-valued functionals like mean-variance and risk measures, to improve forecast comparison and selection.

Contribution

It introduces the concepts of order-sensitivity and equivariance as criteria for choosing scoring functions, particularly for vector-valued functionals, enhancing forecast evaluation methods.

Findings

Order-sensitivity allows comparison of misspecified forecasts.

Equivariant scoring functions respect functional symmetries.

Focus on vector-valued functionals like (mean, variance).

Abstract

The relative performance of competing point forecasts is usually measured in terms of loss or scoring functions. It is widely accepted that these scoring function should be strictly consistent in the sense that the expected score is minimized by the correctly specified forecast for a certain statistical functional such as the mean, median, or a certain risk measure. Thus, strict consistency opens the way to meaningful forecast comparison, but is also important in regression and M-estimation. Usually strictly consistent scoring functions for an elicitable functional are not unique. To give guidance on the choice of a scoring function, this paper introduces two additional quality criteria. Order-sensitivity opens the possibility to compare two deliberately misspecified forecasts given that the forecasts are ordered in a certain sense. On the other hand, equivariant scoring functions obey similar equivariance properties as the functional at hand - such as translation invariance or positive homogeneity. In our study, we consider scoring functions for popular functionals, putting special emphasis on vector-valued functionals, e.g. the pair (mean, variance) or (Value at Risk, Expected Shortfall).