×

Of quantiles and expectiles: consistent scoring functions, Choquet representations and forecast rankings. With discussion and authors’ reply. (English) Zbl 1414.62038

Summary: In the practice of point prediction, it is desirable that forecasters receive a directive in the form of a statistical functional. For example, forecasters might be asked to report the mean or a quantile of their predictive distributions. When evaluating and comparing competing forecasts, it is then critical that the scoring function used for these purposes be consistent for the functional at hand, in the sense that the expected score is minimized when following the directive. We show that any scoring function that is consistent for a quantile or an expectile functional can be represented as a mixture of elementary or extremal scoring functions that form a linearly parameterized family. Scoring functions for the mean value and probability forecasts of binary events constitute important examples. The extremal scoring functions admit appealing economic interpretations of quantiles and expectiles in the context of betting and investment problems. The Choquet-type mixture representations give rise to simple checks of whether a forecast dominates another in the sense that it is preferable under any consistent scoring function. In empirical settings it suffices to compare the average scores for only a finite number of extremal elements. Plots of the average scores with respect to the extremal scoring functions, which we call Murphy diagrams, permit detailed comparisons of the relative merits of competing forecasts.

MSC:

62C05 General considerations in statistical decision theory
62F07 Statistical ranking and selection procedures
62F10 Point estimation
62G05 Nonparametric estimation
91B06 Decision theory
62M20 Inference from stochastic processes and prediction
62-02 Research exposition (monographs, survey articles) pertaining to statistics
PDFBibTeX XMLCite
Full Text: DOI arXiv