Order-Sensitivity and Equivariance of Scoring Functions

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Title: Order-Sensitivity and Equivariance of Scoring Functions
Authors: Fissler, T
Ziegel, JF
Item Type: Working Paper
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).
Issue Date: 28-Nov-2017
URI: http://hdl.handle.net/10044/1/56196
Publisher: arXiv
Copyright Statement: © The Authors
Keywords: math.ST
stat.TH
62C99, 62F07, 62G99, 91B06
Consistency
Decision theory
Elicitability
Equivariance
Homogeneity
M-Estimation
Order-Sensitivity
Point forecasts
Scoring functions
Translation invariance
Publication Status: Published online
Appears in Collections:Mathematics
Statistics
Faculty of Natural Sciences



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