Expectation as a minimizer of the loss function It is a well-known fact that the minimizer of the mean-squared loss (MSE) $$\min\limits_\mu \mathbb{E}_{X} \left(X - \mu \right)^2$$ equals the expectation of $X$.
Are there any alternative non-trivial loss functions $f\left(X, \mu\right)$ such that $$\arg\min_\mu \mathbb{E}_X f\left(X, \mu\right) = \mathbb{E}_X X?$$ And if they exist, what are their pros and cons in comparison with MSE?

Gneiting (2011, JASA):

Subject to weak regularity conditions, a scoring function for a real-valued predictand is consistent for the mean functional if and only if it is a Bregman function, that is, of the form
$$ S(x,y) = \phi(y)-\phi(x)-\phi'(x)(y-x), $$
where $\phi$ is a convex function with subgradient $\phi'$ (Savage, 1971, JASA).

Here $S$ is a scoring function, $x$ is a point prediction, and $y$ is an observation. Gneiting (2011) gives more pointers to literature in his section 3.1.
