# Why scoring functions cannot assess tail properties

## TL;DR

This paper demonstrates that statistical features of distribution tails, such as the extreme value index, cannot be reliably assessed using scoring functions, highlighting limitations in tail property evaluation.

## Contribution

It proves that key tail characteristics are not elicitable and that proper scoring rules cannot distinguish their values, revealing fundamental limitations in tail forecast evaluation.

## Key findings

- Expected scores cannot identify tail properties.
- Max-functionals like the extreme value index are not elicitable.
- Proper scoring rules cannot separate max-functional values.

## Abstract

Motivated by the growing interest in sound forecast evaluation techniques with an emphasis on distribution tails rather than average behaviour, we investigate a fundamental question arising in this context: Can statistical features of distribution tails be elicitable, i.e. be the unique minimizer of an expected score? We demonstrate that expected scores are not suitable to distinguish genuine tail properties in a very strong sense. Specifically, we introduce the class of max-functionals, which contains key characteristics from extreme value theory, for instance the extreme value index. We show that its members fail to be elicitable and that their elicitation complexity is in fact infinite under mild regularity assumptions. Further we prove that, even if the information of a max-functional is reported via the entire distribution function, a proper scoring rule cannot separate max-functional values. These findings highlight the caution needed in forecast evaluation and statistical inference if relevant information is encoded by such functionals.