Measurability of functionals and of ideal point forecasts

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.