Is the mode elicitable relative to unimodal distributions?

Abstract

A statistical functional is said to be 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 is not elicitable if the true distribution can follow any Lebesgue density. We strengthen the result of Heinrich (2014) substantially, showing that the mode is not elicitable if the true distribution can be any strongly unimodal distribution with continuous Lebesgue density, i.e., a continuous density with only one local maximum. Likewise, the mode fails to be identifiable relative to this class.