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. Finally, synthetic examples illustrate that in forecast evaluation Huber loss serves as robust scoring alternative to squared error for expectation point forecasts when some measurements of the realization are faulty.

翻译：划线损失、 其不对称变量及其相关功能( 此处称为“ Huber 函数 ” ) 是在点预测和预测评价的背景下研究的。 分配的划线功能是将预期( 非对称) 划线损失减到最小的一套功能, 是四分位和相应预期损失之间的中间媒介, 也出现在 M- 估计中 。 每个划线功能都是可以取得的, 在概率分布等级的常规性条件薄弱的情况下, 产生一套精确的最小分数, 并完整地描述其一贯的评分功能 。 这些评分功能承认混合代表为基本评分功能的加权平均值 。 每个基本评分可以被解释为在利润和亏损上限的投资决策类别中使用特定预测的相对经济损失 。 最后, 合成例子表明, 在预测评价划线损失时, 划线损失可以作为稳健的评分替代办法, 而不是在对实现结果进行某些计量有误时预测时的标点的标值错误。