  In the practice of point prediction, it is desirable that forecasters receive
a directive in the form of a statistical functional, such as the mean or a
quantile of the predictive distribution. 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, respectively, can be represented as a mixture of 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 quantile and expectile functionals along with the
respective extremal scoring functions admit appealing economic interpretations
in terms of thresholds in decision making.
  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.
