I provide an axiomatic characterization of the class of location parameters derived from well-defined convex minimization problems. The resulting class includes a broad set of estimators, such as the mean and —- in the limit —- the median, but excludes the mode.
I also address the problem of comparing location parameters within this set by reframing the optimization problem as the one of a decision maker selecting a point forecast to minimize an expected loss. I formalize the decision maker’s preferences over two key attributes of the resulting estimator: robustness (sensitivity to outliers) and directionality (asymmetry in weighting positive vs. negative deviations), and I discuss how they depend on the shape of the loss function. While a direct comparison of estimators
can be difficult because of the underlying trade-off between robustness and directionality, I show that, under mild conditions, differences between estimators can be fully reinterpreted as differences in directionality only.