  We study stochastic programs where the decision-maker cannot observe the
distribution of the exogenous uncertainties but has access to a finite set of
independent samples from this distribution. In this setting, the goal is to
find a procedure that transforms the data to an estimate of the expected cost
function under the unknown data-generating distribution, i.e., a predictor, and
an optimizer of the estimated cost function that serves as a near-optimal
candidate decision, i.e., a prescriptor. As functions of the data, predictors
and prescriptors constitute statistical estimators. We propose a
meta-optimization problem to find the least conservative predictors and
prescriptors subject to constraints on their out-of-sample disappointment. The
out-of-sample disappointment quantifies the probability that the actual
expected cost of the candidate decision under the unknown true distribution
exceeds its predicted cost. Leveraging tools from large deviations theory, we
prove that this meta-optimization problem admits a unique solution: The best
predictor-prescriptor pair is obtained by solving a distributionally robust
optimization problem over all distributions within a given relative entropy
distance from the empirical distribution of the data.
