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

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.

Estimation of Tail Risk Based on Extreme Expectiles

“…Also, Ehm et al . () have shown that expectiles are optimal decision thresholds in certain binary investment problems. Another motivation for the adoption of expectiles in risk management, according to Taylor (), is that they are very closely related to the classical mean and the popular ES.…”

“…Bellini and Di Bernardino (2017) provide a transparent financial meaning of expectiles in terms of the gain-loss ratio, which is a popular performance measure in portfolio management and is well known in the literature on no-gooddeal valuation in incomplete markets (see Bellini and Di Bernardino (2017) and references therein). Also, Ehm et al (2016) have shown that expectiles are optimal decision thresholds in certain binary investment problems. Another motivation for the adoption of expectiles in risk management, according to Taylor (2008), is that they are very closely related to the classical mean and the popular ES.…”

Enforcing calibration in ensemble postprocessing

“…Value Score plots are examples of what Ehm et al . () have called ‘Murphy diagrams’, in recognition of the work of Murphy ().…”

“…(13) as a function of C/L, and so represent forecast value across a spectrum of decision makers and their decision problems. Value Score plots are examples of what Ehm et al (2016) have called 'Murphy diagrams', in recognition of the work of Murphy (1977). Figure 6 shows Value Score curves for probability forecasts of the event T min ≤ 0 • C, that have been derived from the minimumtemperature forecasts postprocessed using the VSV algorithm, Eq.…”

Monitoring trends in ensemble forecast performance focusing on surface variables and high‐impact events

“…The basic building element of proper scoring rules is the elementary score (Ehm et al . ; Ben Bouallègue et al . ).…”