  While solutions of Distributionally Robust Optimization (DRO) problems can
sometimes have a higher out-of-sample expected reward than the Sample Average
Approximation (SAA), there is no guarantee. In this paper, we introduce a class
of Distributionally Optimistic Optimization (DOO) models, and show that it is
always possible to ``beat" SAA out-of-sample if we consider not just worst-case
(DRO) models but also best-case (DOO) ones. We also show, however, that this
comes at a cost: Optimistic solutions are more sensitive to model error than
either worst-case or SAA optimizers, and hence are less robust and calibrating
the worst- or best-case model to outperform SAA may be difficult when data is
limited.
