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Measurability of functionals and of ideal point forecasts

Author

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  • Tobias Fissler
  • Hajo Holzmann
Abstract
The ideal probabilistic forecast for a random variable $Y$ based on an information set $\mathcal{F}$ is the conditional distribution of $Y$ given $\mathcal{F}$. 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 $\mathcal{F}$-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.

Suggested Citation

  • Tobias Fissler & Hajo Holzmann, 2022. "Measurability of functionals and of ideal point forecasts," Papers 2203.08635, arXiv.org.
    • Handle: RePEc:arx:papers:2203.08635
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    File URL: http://arxiv.org/pdf/2203.08635
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    References listed on IDEAS

    as
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    5. Tobias Fissler & Silvana M. Pesenti, 2022. "Sensitivity Measures Based on Scoring Functions," Papers 2203.00460, arXiv.org, revised Jul 2022.
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    Cited by:

    1. Tobias Fissler & Silvana M. Pesenti, 2022. "Sensitivity Measures Based on Scoring Functions," Papers 2203.00460, arXiv.org, revised Jul 2022.
    2. Qiuqi Wang & Ruodu Wang & Johanna Ziegel, 2022. "E-backtesting," Papers 2209.00991, arXiv.org, revised Dec 2024.
    3. Fissler, Tobias & Pesenti, Silvana M., 2023. "Sensitivity measures based on scoring functions," European Journal of Operational Research, Elsevier, vol. 307(3), pages 1408-1423.

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