Is the mode elicitable relative to unimodal distributions?

TL;DR

This paper investigates the elicitability of the mode in statistical distributions, proving it cannot be elicited or identified when the true distribution has a continuous density with a unique local maximum.

Contribution

It extends previous results by showing the mode's non-elicitability and non-identifiability for a broader class of distributions with continuous densities and a single local maximum.

Findings

Mode is not elicitable for distributions with continuous densities and a unique local maximum.

Mode cannot be identified relative to this class of distributions.

Elicitation of the mode is fundamentally limited under these conditions.

Abstract

Statistical functionals are called elicitable if there exists a loss or scoring function under which the functional is the optimal point forecast in expectation. While the mean and quantiles are elicitable, it has been shown in Heinrich (2014) that the mode cannot be elicited if the true distribution can follow any Lebesgue density. We strengthen this result substantially, showing that the mode cannot be elicited if the true distribution is any distribution with continuous Lebesgue density and unique local maximum. Likewise, the mode fails to be identifiable relative to this class.