We consider an investor who has sold a contingent claim and intends to minimize the maximal expected weighted shortfall. Here, the maximum is taken over a family of models. We call the associated minimizing strategy robust-efficient. The problem to determine a robust-efficient strategy is closely related to the
statistical problem of testing a composite hypothesis against a composite alternative. The solution to this statistical testing problem is provided on a general level by means of a
least-favorable pair. We apply these results to derive the robust-efficient strategy for a class of Binomial-models with uncertain transition probabilities and for a class of generalized
Black-Scholes models where volatility is subject to a random jump with uncertain mean and variance.