Name of mean absolute error analogue to Brier score?

Yesterday’s question Determine accuracy of model which estimates probability of event got me curious about probability scoring.

The Brier score
$$\frac{1}{N}\sum\limits _{i=1}^{N}(\text{prediction}_i – \text{reference}_i)^2$$
is a mean squared error measure.
Does the analogous mean absolute error performance measure
$$\frac{1}{N}\sum\limits _{i=1}^{N}|\text{prediction}_i – \text{reference}_i|$$
have a name, too?

Answer

Answer seems to be: no, because MAE doesn’t lead to a proper scoring rule.

See Loss Functions for Binary Class Probability Estimation and Classification: Structure and Applications where the MAE is discussed under “Counterexamples of proper scoring rules”.

Attribution
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