## 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.

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.

Original language | English |
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DOIs | |

Publication status | Published - 2022 |

## Austrian Classification of Fields of Science and Technology (ÖFOS)

- 101029 Mathematical statistics