Higher order elicitability and Osband’s principle

Fissler, Tobias; Ziegel, Johanna F. (2016). Higher order elicitability and Osband’s principle. Annals of statistics, 44(4), pp. 1680-1707. Institute of Mathematical Statistics 10.1214/16-AOS1439

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A statistical functional, such as the mean or the median, is called elicitable if there is a scoring function or loss function such that the correct forecast of the functional is the unique minimizer of the expected score. Such scoring functions are called strictly consistent for the functional. The elicitability of a functional opens the possibility to compare competing forecasts and to rank them in terms of their realized scores. In this paper, we explore the notion of elicitability for multi-dimensional functionals and give both necessary and sufficient conditions for strictly consistent scoring functions. We cover the case of functionals with elicitable components, but we also show that one-dimensional functionals that are not elicitable can be a component
of a higher order elicitable functional. In the case of the variance, this is a
known result. However, an important result of this paper is that spectral risk
measures with a spectral measure with finite support are jointly elicitable if
one adds the “correct” quantiles. A direct consequence of applied interest is
that the pair (Value at Risk, Expected Shortfall) is jointly elicitable under mild
conditions that are usually fulfilled in risk management applications.

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