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Abstract
We introduce a theoretical framework of elicitability and identifiability of setvalued functionals, such as quantiles, prediction intervals, and systemic risk measures. A functional is elicitable if it is the unique minimiser of an expected scoring function, and identifiable if it is the unique zero of an expected identification function; both notions are essential for forecast ranking and validation, and M and Zestimation. Our framework distinguishes between exhaustive forecasts, being setvalued and aiming at correctly specifying the entire functional, and selective forecasts, content with solely specifying a single point in the correct functional. We establish a mutual exclusivity result: A setvalued functional can be either selectively elicitable or exhaustively elicitable or not elicitable at all. Notably, since quantiles are well known to be selectively elicitable, they fail to be exhaustively elicitable. We further show that the classes of prediction intervals and Vorob’ev quantiles turn out to be exhaustively elicitable, hence not selectively elicitable, but still selectively identifiable. In particular, we provide a mixture representation of elementary exhaustive scores, leading the way to Murphy diagrams. We establish that the shortest prediction interval and those specified by an endpoint or midpoint in general fail to be elicitable with respect to either notion, unless an endpoint is given via a quantile. We end with a comprehensive literature review on common practice in forecast evaluation of setvalued functionals.
Originalsprache  Englisch 

Seiten (von  bis)  1034  1084 
Fachzeitschrift  Electronic Journal of Statistics 
Jahrgang  15 
Ausgabenummer  1 
DOIs  
Publikationsstatus  Veröffentlicht  2021 
Österreichische Systematik der Wissenschaftszweige (ÖFOS)
 101029 Mathematische Statistik
Projekte
 1 Abgeschlossen

Dynamic measures of systemic risk
Rudloff, B., Diem, C. & Johnson, K.
1/03/19 → 28/02/23
Projekt: Forschungsförderung