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Measurability of functionals and of ideal point forecasts

16 March 2022
Tobias Fissler
H. Holzmann
ArXiv (abs)PDFHTML
Abstract

The ideal probabilistic forecast for a random variable YYY based on an information set F\mathcal{F}F is the conditional distribution of YYY given F\mathcal{F}F. In the context of point forecasts aiming to specify a functional TTT 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\mathcal{F}F-measurable random variable. To that end, the appropriate notion of measurability of TTT is clarified and this measurability is established for a large class of practically relevant functionals, including elicitable ones. More generally, the measurability of TTT implies the measurability of any point forecast which arises by applying TTT to a probabilistic forecast. Similar measurability results are established for proper scoring rules, the main tool to evaluate the predictive accuracy of probabilistic forecasts.

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