Sensitivity Measures Based on Scoring Functions

38 Pages Posted: 28 Mar 2022 Last revised: 24 Oct 2022

See all articles by Tobias Fissler

Tobias Fissler

Vienna University of Economics and Business

Silvana M. Pesenti

University of Toronto

Date Written: July 1, 2022


We propose a holistic framework for constructing sensitivity measures for any elicitable functional T of a response variable. The sensitivity measures, termed score-based sensitivities, are constructed via scoring functions that are (strictly) consistent for T. These score-based sensitivities quantify the relative improvement in predictive accuracy when available information, e.g., from explanatory variables, is used ideally. We establish intuitive and desirable properties of these sensitivities and discuss advantageous choices of scoring functions leading to scale-invariant sensitivities.

Since elicitable functionals typically possess rich classes of (strictly) consistent scoring functions, we demonstrate how Murphy diagrams can provide a picture of all score-based sensitivity measures. We discuss the family of score-based sensitivities for the mean functional (of which the Sobol indices are a special case) and risk functionals such as Value-at-Risk, and the pair Value-at-Risk and Expected Shortfall. The sensitivity measures are illustrated using numerous examples, including the Ishigami--Homma test function. In a simulation study, estimation of score-based sensitivities for a non-linear insurance portfolio is performed using neural nets.

Link to the published version:

Keywords: Consistency; Elicitability; Expected Shortfall; Information value; Value-at-Risk

JEL Classification: C52, C44, G22, C14

Suggested Citation

Fissler, Tobias and Pesenti, Silvana M., Sensitivity Measures Based on Scoring Functions (July 1, 2022). Available at SSRN: or

Tobias Fissler (Contact Author)

Vienna University of Economics and Business ( email )

Welthandelsplatz 1
Vienna, Wien 1020

Silvana M. Pesenti

University of Toronto ( email )

700 University Avenue 9F
Toronto, Ontario

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