Evaluating probabilities: asymmetric scoring rules
Management Science
Coherence and Consistency of Investors' Probability Judgments
Management Science
Probability Elicitation, Scoring Rules, and Competition Among Forecasters
Management Science
Quantile-Parameterized Distributions
Decision Analysis
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Quantile assessments are commonly encountered in the elicitation of probability distributions in decision analysis, forecasting, and risk analysis. Scoring rules have been developed to provide ex ante incentives for careful and truthful assessments and ex post evaluation measures in the context of probability assessment. We show that these scoring rules designed for probability assessment provide inappropriate incentives if used for quantile assessment. We investigate the properties of a linear family of scoring rules that are intended specifically for quantile assessment (including the assessment of multiple quantiles) and can be related to a realistic decision-making problem. These rules provide proper incentives for quantile assessment and yield higher expected scores for distributions that are more informative in the sense of having less dispersion. We discuss the special case of interval forecasts and a generalization involving transformations, and we briefly mention other possible extensions.