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.
Acknowledgments
The author would like to thank Jonas Brehmer, Professor Tilmann Gneiting, Deryn Griffiths, Robert Fawcett, Nicholas Loveday and two anonymous reviewers for the constructive comments and suggestions, which improved the quality of this manuscript. I also thank my family for their support. Finally, an expression of gratitude to Harry Jack, who introduced me to Huber loss, and to Michael Foley, who four years ago made the decision to start scoring Bureau of Meteorology temperature forecasts using Huber loss. I would not have otherwise embarked on this study.
Citation
Robert J. Taggart. "Point forecasting and forecast evaluation with generalized Huber loss." Electron. J. Statist. 16 (1) 201 - 231, 2022. https://doi.org/10.1214/21-EJS1957
Information
