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

TL;DR

This paper explores the theoretical foundations of scoring functions for quantiles and expectiles, providing mixture representations, economic interpretations, and practical tools like Murphy diagrams for forecast evaluation.

Contribution

It introduces a unified framework for consistent scoring functions via Choquet mixtures and develops Murphy diagrams for practical forecast comparison.

Findings

Scoring functions can be represented as mixtures of extremal functions.

Murphy diagrams enable comprehensive forecast comparisons.

The framework applies to quantiles, expectiles, and other functionals.

Abstract

In the practice of point prediction, it is desirable that forecasters receive a directive in the form of a statistical functional, such as the mean or a quantile of the predictive distribution. 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 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 quantile and expectile functionals along with the respective extremal scoring functions admit appealing economic interpretations in terms of thresholds in decision making. 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.