{"title":"Measurability of functionals and of ideal point forecasts","authors":"Tobias Fissler, H. Holzmann","doi":"10.1214/22-EJS2062","DOIUrl":null,"url":null,"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.","PeriodicalId":49272,"journal":{"name":"Electronic Journal of Statistics","volume":" ","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Electronic Journal of Statistics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1214/22-EJS2062","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
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.
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.