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In the practice of point prediction, it is desirable that forecasters receivea directive in the form of a statistical functional, such as the mean or aquantile of the predictive distribution. When evaluating and comparingcompeting forecasts, it is then critical that the scoring function used forthese purposes be consistent for the functional at hand, in the sense that theexpected score is minimized when following the directive.  We show that any scoring function that is consistent for a quantile or anexpectile functional, respectively, can be represented as a mixture of extremalscoring functions that form a linearly parameterized family. Scoring functionsfor the mean value and probability forecasts of binary events constituteimportant examples. The quantile and expectile functionals along with therespective extremal scoring functions admit appealing economic interpretationsin terms of thresholds in decision making.  The Choquet type mixture representations give rise to simple checks ofwhether a forecast dominates another in the sense that it is preferable underany consistent scoring function. In empirical settings it suffices to comparethe average scores for only a finite number of extremal elements. Plots of theaverage scores with respect to the extremal scoring functions, which we callMurphy diagrams, permit detailed comparisons of the relative merits ofcompeting forecasts.

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