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).
Highlights
From the cradle to the grave, human life is full of decisions
We focus on point forecasts that may be vector-valued, which is why we assume A ⊆ Rk for some k ≥ 1 and we equip the Borel set A with the Borel σ-algebra
We investigate to which extent invariance or equivariance properties of elicitable functionals are reflected in their respective consistent scoring functions
Summary
From the cradle to the grave, human life is full of decisions. Due to the inherent nature of time, decisions have to be made today, but at the same time, they are supposed to account for unknown and uncertain future events. We discuss their connections (Lemma 3.5) and give conditions when such scoring functions exist (Lemma B.2, Propositions 3.7, 3.8, Corollary 3.16) and of what form they are for the most relevant functionals, such as vectors of quantiles (Propositions 3.11, 3.12, Example 3.14), expectiles (Proposition 3.15), ratios of expectations (Propositions 3.6, 3.9, 3.10, 3.17), the pair of mean and variance (Proposition 3.18, Example 3.19), and the pair consisting of Value at Risk and Expected Shortfall (Proposition 3.20, Example 3.21), two important risk measures in banking and insurance. Appendix B consists of technical results, while all proofs are of the main part of this paper are deferred to Appendix C
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