Conditional Quantiles: An Operator-Theoretical Approach

Castro, Luciano I. de; Costa, Bruno; Galvao, Antônio F.; Zubelli, Jorge P.

doi:10.2139/ssrn.3924597

Cited by 2 publications

(2 citation statements)

References 20 publications

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“…Therefore, the ideal forecast q α (Y |F) is a random closed set in the sense of Definition 1.1.1 in Molchanov (2017). This complements the measurability argument of the quantile provided recently in de Castro et al (2021).…”

Section: Measurability Of Functionalssupporting

confidence: 83%

Measurability of functionals and of ideal point forecasts

Fissler¹,

Holzmann²

2022

Preprint

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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.

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Section: Measurability Of Functionalssupporting

confidence: 83%

Measurability of functionals and of ideal point forecasts

Fissler¹,

Holzmann²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…where X and X are independent copies with distribution F . From both (8) and ( 7) one can readily see that CRPS(F, y) is continuous as a function of y, invoking the dominated convergence theorem. Further, for fixed y, (8) implies measurability of CRPS(F, y) as a map in F , while from (7) To apply Lemma 14, observe that LOGS(p, y) is right-continuous as a function of y by our assumption on p and continuity of the logarithm.…”

Section: Measurability Of Scoring Rulesmentioning

confidence: 85%

Measurability of functionals and of ideal point forecasts

Fissler

Holzmann

2022

Electron. J. Statist.

View full text Add to dashboard Cite

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.

show abstract

Conditional Quantiles: An Operator-Theoretical Approach

Cited by 2 publications

References 20 publications

Measurability of functionals and of ideal point forecasts

Measurability of functionals and of ideal point forecasts

Measurability of functionals and of ideal point forecasts

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