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B1751
Title: Of quantiles and expectiles: Consistent scoring functions, mixture representations, and forecast rankings Authors:  Werner Ehm - Heidelberg Institute for Theoretical Studies (Germany)
Alexander Jordan - HITS gGmbH, Heidelberg Institute for Theoretical Studies (Germany)
Fabian Krueger - Heidelberg Instititute for Theoretical Studies gGmbH (Germany)
Tilmann Gneiting - University of Heidelberg (Germany) [presenting]
Abstract: 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, respectively, 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 economic interpretations in terms of betting and investment problems.The mixture representations allow for simple checks of whether a forecast dominates another, in the sense that it is preferable under any consistent scoring function. 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.