Of Quantiles and Expectiles: Consistent Scoring Functions, Choquet\n Representations, and Forecast Rankings

In the practice of point prediction, it is desirable that forecasters receive\na directive in the form of a statistical functional, such as the mean or a\nquantile of the predictive distribution. When evaluating and comparing\ncompeting forecasts, it is then critical that the scoring function used for\nthese purposes be consistent for the functional at hand, in the sense that the\nexpected score is minimized when following the directive.\n We show that any scoring function that is consistent for a quantile or an\nexpectile functional, respectively, can be represented as a mixture of extremal\nscoring functions that form a linearly parameterized family. Scoring functions\nfor the mean value and probability forecasts of binary events constitute\nimportant examples. The quantile and expectile functionals along with the\nrespective extremal scoring functions admit appealing economic interpretations\nin terms of thresholds in decision making.\n The Choquet type mixture representations give rise to simple checks of\nwhether a forecast dominates another in the sense that it is preferable under\nany consistent scoring function. In empirical settings it suffices to compare\nthe average scores for only a finite number of extremal elements. Plots of the\naverage scores with respect to the extremal scoring functions, which we call\nMurphy diagrams, permit detailed comparisons of the relative merits of\ncompeting forecasts.\n