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.

Sensitivity Measures Based on Scoring Functions

“…and Holzmann (2022) for measurability results for functionals of interest such as the mean, expectiles, Value-at-Risk (VaR), and Expected Shortfall (ES).For a random variable Y in some class Y ⊆ L 0 (Ω, F, P) we denote its cumulative distribution function by F Y (•) = P(Y ≤ •). For a sub-σ-algebra A ⊆ F -often referred to as information set -we denote by F Y |A a regular version of the conditional distribution of Y given A.…”

E-backtesting

“…Let r t and z t be forecasts for ρ(X t |F t−1 ) and φ(X t |F t−1 ) made at time t − 1, respectively. Note that ρ(X t |F t−1 ) and φ(X t |F t−1 ) are themselves random variables and F t−1 -measurable for all relevant functionals of interest (Fissler and Holzmann, 2022).…”