Of quantiles and expectiles: consistent scoring functions, Choquet representations and forecast rankings. (June 2016)

• Record Type: Journal Article
• Authors: Ehm, Werner; Gneiting, Tilmann; Jordan, Alexander; Krüger, Fabian
• Is Part Of: Journal of the Royal Statistical Society. Volume 78:Number 3(2016:Jun.)
• Page Start: 505
• Page End: 562
• Publication Date: 2016-06
• Subjects: Choquet representation -- Consistent scoring function -- Decision theory -- Economic utility -- Elicitable -- Expectile -- Forecast ranking -- Order sensitivity -- Point forecast -- Probability forecast -- Quantile; Statistics -- Periodicals; Great Britain -- Statistics -- Periodicals; 519.2
• DOI: 10.1111/rssb.12154
• Languages: English
• ISSNs: 1369-7412

Abstract

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.