Quantile Evaluation, Sensitivity to Bracketing, and Sharing Business Payoffs

Abstract

From forecasting competitions to conditional value-at-risk requirements, the use of multiple quantile assessments is growing in practice. To evaluate them, we use a rule from the general class of proper scoring rules for a forecaster's multiple quantiles of a single uncertain quantity of interest. The general rule is additive in the component scores. Each component contains a function that measures its quantile's distance from the realization and weights its contribution to the overall score. To determine this function, we propose that the score of a group's combined quantile should be better than that of a randomly-selected forecaster's quantile only when the forecasters bracket the realization, or their quantiles do not all fall on the same side of the realization. If a score satisfies this property, we say it is sensitive to bracketing. We characterize the class of proper scoring rules that is sensitive to bracketing when the decision maker uses a generalized average to combine forecasters' quantiles. Finally, we show how weights can be set to match the payoffs in many important business contexts.

Keywords: quantile forecasts; forecast evaluation, probability elicitation, proper scoring rules, expert aggregation, expert combination