Order-sensitivity and equivariance of scoring functions

Fissler, Tobias; Ziegel, Johanna F. (2019). Order-sensitivity and equivariance of scoring functions. Electronic journal of statistics, 13(1), pp. 1166-1211. Institute of Mathematical Statistics 10.1214/19-EJS1552

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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 acertain 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 notunique. To give guidance on the choice of a scoring function, this paper introduces two additional quality criteria. Order-sensitivity opens the pos-sibility 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 (Valueat Risk, Expected Shortfall).

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