Is Pitman Closeness a Reasonable Criterion?

The criterion of Pitman closeness has been proposed as an alternative comparison criterion to quadratic losses and, more generally, to decision theory. However, it may lead to quite paradoxical phenomena, the most dramatic being a possible nontransitivity. The criterion takes into consideration the joint distribution of the compared estimators, but this consideration may be misleading in the selection of the “best” estimator. We show through examples that this criterion is not consistent with a decision theoretic analysis and that it should be used very cautiously, if ever.