Point forecasting and forecast evaluation with generalized Huber loss

TL;DR

This paper explores the properties of Huber loss and its variants in point forecasting, providing a theoretical framework for forecast evaluation and highlighting their practical relevance in weather forecast assessment.

Contribution

It introduces the concept of Huber functionals, characterizes their elicitable scoring functions, and connects these to economic loss interpretations for forecast evaluation.

Findings

Huber functionals are intermediates between quantiles and expectiles.

Elicitable scoring functions for Huber functionals are characterized and represented as mixtures.

The theory applies to weather forecast assessment and economic decision-making.

Abstract

Huber loss, its asymmetric variants and their associated functionals (here named Huber functionals) are studied in the context of point forecasting and forecast evaluation. The Huber functional of a distribution is the set of minimizers of the expected (asymmetric) Huber loss, is an intermediary between a quantile and corresponding expectile, and also arises in M-estimation. Each Huber functional is elicitable, generating the precise set of minimizers of an expected score, subject to weak regularity conditions on the class of probability distributions, and has a complete characterization of its consistent scoring functions. Such scoring functions admit a mixture representation as a weighted average of elementary scoring functions. Each elementary score can be interpreted as the relative economic loss of using a particular forecast for a class of investment decisions where profits and losses are capped. The relevance of this theory for comparative assessment of weather forecasts is also discussed.