Measurability of functionals and of ideal point forecasts

IF 1 4区 数学 Q3 STATISTICS & PROBABILITY
Tobias Fissler, H. Holzmann
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引用次数: 4

Abstract

. The ideal probabilistic forecast for a random variable Y based on an information set F is the conditional distribution of Y given F . In the context of point forecasts aiming to specify a functional T such as the mean, a quantile or a risk measure, the ideal point forecast is the respective functional applied to the conditional distribution. This paper provides a theoretical justification why this ideal forecast is actually a forecast, that is, an F -measurable random variable. To that end, the appropriate notion of measurability of T is clarified and this measurability is established for a large class of practically relevant functionals, including elicitable ones. More generally, the measurability of T implies the measurability of any point forecast which arises by applying T to a probabilistic forecast. Similar measurability results are established for proper scoring rules, the main tool to evaluate the predictive accuracy of probabilistic forecasts.
函数和理想点预测的可测量性
. 基于信息集F的随机变量Y的理想概率预测是给定F的Y的条件分布。在旨在指定函数T(如平均值、分位数或风险度量)的点预测环境中,理想的点预测是应用于条件分布的各自函数。本文提供了一个理论证明,为什么这个理想的预测实际上是一个预测,即F可测量的随机变量。为此,澄清了T的可测量性的适当概念,并为一大类实际相关的泛函(包括可引出的泛函)建立了这种可测量性。更一般地说,T的可测量性意味着将T应用于概率预测而产生的任何点预测的可测量性。适当的评分规则是评估概率预测准确性的主要工具,建立了相似的可测量性结果。
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来源期刊
Electronic Journal of Statistics
Electronic Journal of Statistics STATISTICS & PROBABILITY-
CiteScore
1.80
自引率
9.10%
发文量
100
审稿时长
3 months
期刊介绍: The Electronic Journal of Statistics (EJS) publishes research articles and short notes on theoretical, computational and applied statistics. The journal is open access. Articles are refereed and are held to the same standard as articles in other IMS journals. Articles become publicly available shortly after they are accepted.
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